{
 "cells": [
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "# Monotoniczność funkcji",
   "id": "6d71d6991752e211"
  },
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": [
    "```{admonition} Wskazówka\n",
    ":class: tip\n",
    "\n",
    "[![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/DyonOylloug/generator_zadan/HEAD?labpath=docs%2Fmonotonicznosc.ipynb) <- interaktywna sesja notebooka\n",
    "\n",
    "```"
   ],
   "id": "80964cf5560052af"
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-05-09T14:21:50.056799Z",
     "start_time": "2025-05-09T14:21:47.253264Z"
    }
   },
   "cell_type": "code",
   "source": [
    "from random import choice\n",
    "\n",
    "from IPython.display import display, Image, Markdown, Latex\n",
    "\n",
    "import generator_zadan.generatory as gz\n",
    "\n",
    "print(gz.__version__)"
   ],
   "id": "b9027f9e730828a6",
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2.11\n"
     ]
    }
   ],
   "execution_count": 1
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-05-09T14:22:08.719742Z",
     "start_time": "2025-05-09T14:22:08.716686Z"
    }
   },
   "cell_type": "code",
   "source": "ile_zadan_przykladowych = 10",
   "id": "558474f70b4a5334",
   "outputs": [],
   "execution_count": 5
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-05-09T14:22:10.338312Z",
     "start_time": "2025-05-09T14:22:09.978268Z"
    }
   },
   "cell_type": "code",
   "source": [
    "zadanie = gz.monotonicznosc(choice([1,2,3]))\n",
    "zadanie"
   ],
   "id": "c6d3abf5b4cda7cf",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('Zbadać monotoniczność i ekstrema funkcji\\n\\t\\\\[\\n\\t\\tf(x)=\\\\frac{2 x^{3} + 2 x - 2}{x - 1} \\n\\t\\\\]\\n',\n",
       " \"$D_f\\\\colon \\\\mathbb{R}\\\\setminus \\\\left\\\\{1\\\\right\\\\}$\\\\newline\\n$f'(x)=\\\\frac{2 x^{2} \\\\left(2 x - 3\\\\right)}{\\\\left(x - 1\\\\right)^{2}}$\\\\newline\\n$f'(x) > 0 \\\\textnormal{ dla: }\\\\frac{3}{2} < x \\\\implies f(x) \\\\textnormal{ jest rosnąca dla } x \\\\in \\\\left(\\\\frac{3}{2},\\\\infty\\\\right) $\\\\newline\\n$f'(x) < 0 \\\\textnormal{ dla: }\\\\left(0 < x \\\\wedge x < 1\\\\right) \\\\vee \\\\left(1 < x \\\\wedge x < \\\\frac{3}{2}\\\\right) \\\\vee x < 0 \\\\implies f(x) \\\\textnormal{ jest malejąca dla } x \\\\in \\\\left(-\\\\infty,1\\\\right) \\\\textnormal{ oraz } x \\\\in  \\\\left(1,\\\\frac{3}{2}\\\\right) $\\\\newline\\n$f'(x) = 0 \\\\textnormal{ dla } x \\\\in \\\\left\\\\{ 0, \\\\  \\\\frac{3}{2}\\\\right\\\\} \\\\implies f_{\\\\min}\\\\left(\\\\frac{3}{2}\\\\right) = \\\\frac{31}{2}$\")"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 6
  },
  {
   "metadata": {
    "tags": [
     "hide-input"
    ],
    "ExecuteTime": {
     "end_time": "2025-05-09T14:22:11.208531Z",
     "start_time": "2025-05-09T14:22:11.205232Z"
    }
   },
   "cell_type": "code",
   "source": [
    "print(\"\\033[34m** Zadanie **\" + '*' * 81 + '\\033[0m')\n",
    "print(zadanie[0])\n",
    "print(\"\\033[34m*\\033[0m\" * 95)\n",
    "print(\"\\033[32m** Rozwiązanie **\" + '*' * 78 + '\\033[0m')\n",
    "print(zadanie[1])\n",
    "print(\"\\033[32m*\\033[0m\" * 95)\n",
    "print(zadanie[1].split('$')[3].split('\\\\implies'))\n"
   ],
   "id": "c13b9487ff9ef145",
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m** Zadanie ***********************************************************************************\u001B[0m\n",
      "Zbadać monotoniczność i ekstrema funkcji\n",
      "\t\\[\n",
      "\t\tf(x)=\\frac{2 x^{3} + 2 x - 2}{x - 1} \n",
      "\t\\]\n",
      "\n",
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n",
      "$D_f\\colon \\mathbb{R}\\setminus \\left\\{1\\right\\}$\\newline\n",
      "$f'(x)=\\frac{2 x^{2} \\left(2 x - 3\\right)}{\\left(x - 1\\right)^{2}}$\\newline\n",
      "$f'(x) > 0 \\textnormal{ dla: }\\frac{3}{2} < x \\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(\\frac{3}{2},\\infty\\right) $\\newline\n",
      "$f'(x) < 0 \\textnormal{ dla: }\\left(0 < x \\wedge x < 1\\right) \\vee \\left(1 < x \\wedge x < \\frac{3}{2}\\right) \\vee x < 0 \\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left(-\\infty,1\\right) \\textnormal{ oraz } x \\in  \\left(1,\\frac{3}{2}\\right) $\\newline\n",
      "$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\{ 0, \\  \\frac{3}{2}\\right\\} \\implies f_{\\min}\\left(\\frac{3}{2}\\right) = \\frac{31}{2}$\n",
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "[\"f'(x)=\\\\frac{2 x^{2} \\\\left(2 x - 3\\\\right)}{\\\\left(x - 1\\\\right)^{2}}\"]\n"
     ]
    }
   ],
   "execution_count": 7
  },
  {
   "metadata": {
    "tags": [
     "hide-input"
    ],
    "ExecuteTime": {
     "end_time": "2025-05-09T14:23:10.589840Z",
     "start_time": "2025-05-09T14:22:13.146165Z"
    }
   },
   "cell_type": "code",
   "source": [
    "for i in range(1, ile_zadan_przykladowych + 1):\n",
    "    zadanie = gz.monotonicznosc(choice([1,2,3]))\n",
    "    print(f\"\\033[34m** Zadanie {i} **\" + '*' * 80 + '\\033[0m')\n",
    "    display(Markdown(zadanie[0].split('\\n\\t')[0]))\n",
    "    display(Latex('$$' + (zadanie[0].split('\\n\\t')[2]) + '$$'))\n",
    "    print(\"\\033[34m*\\033[0m\" * 95)\n",
    "    print(\"\\033[32m** Rozwiązanie **\" + '*' * 78 + '\\033[0m')\n",
    "    display(Markdown('$$' + zadanie[1].split('$')[1].replace('\\\\left\\\\{', '\\\\left\\\\lbrace').replace('\\\\right\\\\}', '\\\\right\\\\rbrace') + '$$'))\n",
    "    display(Markdown('$$' + zadanie[1].split('$')[5].split('\\\\implies')[0] + '\\\\implies' + '$$' ))\n",
    "    display(Markdown('$$' + '\\\\implies' + zadanie[1].split('$')[5].split('\\\\implies')[1] + '$$' ))\n",
    "    display(Markdown('$$' + zadanie[1].split('$')[7].split('\\\\implies')[0] + '\\\\implies' + '$$' ))\n",
    "    display(Markdown('$$' + '\\\\implies' + zadanie[1].split('$')[7].split('\\\\implies')[1] + '$$' ))\n",
    "    display(Markdown('$$' + zadanie[1].split('$')[9].replace('\\\\left\\\\{', '\\\\left\\\\lbrace').replace('\\\\right\\\\}', '\\\\right\\\\rbrace') + '$$'))\n",
    "    print(\"\\033[32m*\\033[0m\" * 95)"
   ],
   "id": "2b94ffcfcafd7d93",
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m** Zadanie 1 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{- x^{2} - x + 1}{2 x - 2} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace1\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x > 0 \\wedge x < 2 \\wedge x \\neq 1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(0,1  \\right)\\textnormal{ oraz } x \\in \\left(1,2  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }2 < x \\vee x < 0 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( -\\infty, 0  \\right)\\textnormal{ oraz } x \\in \\left(2, \\infty  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace 0, \\  2\\right\\rbrace \\implies f_{\\min}\\left(0\\right) = - \\frac{1}{2} \\textnormal{ oraz } f_{\\max}\\left(2\\right) = - \\frac{5}{2}$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 2 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{x^{2} + x + 2}{1 - x} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace1\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x > -1 \\wedge x < 3 \\wedge x \\neq 1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(-1,1  \\right)\\textnormal{ oraz } x \\in \\left(1,3  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }3 < x \\vee x < -1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( -\\infty, -1  \\right)\\textnormal{ oraz } x \\in \\left(3, \\infty  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace -1, \\  3\\right\\rbrace \\implies f_{\\min}\\left(-1\\right) = 1 \\textnormal{ oraz } f_{\\max}\\left(3\\right) = -7$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 3 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{3 x^{3}}{- x - 1} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace-1\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x < - \\frac{3}{2} \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(-\\infty,- \\frac{3}{2}\\right)  $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }\\left(- \\frac{3}{2} < x \\wedge x < -1\\right) \\vee \\left(-1 < x \\wedge x < 0\\right) \\vee 0 < x \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left(- \\frac{3}{2},-1\\right) \\textnormal{ oraz } x \\in  \\left(-1,\\infty\\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace - \\frac{3}{2}, \\  0\\right\\rbrace \\implies f_{\\max}\\left(- \\frac{3}{2}\\right) = - \\frac{81}{4}$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 4 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{4 x^{2} + 3}{- 2 x - 1} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace- \\frac{1}{2}\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x > - \\frac{3}{2} \\wedge x < \\frac{1}{2} \\wedge x \\neq - \\frac{1}{2} \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(- \\frac{3}{2},- \\frac{1}{2}  \\right)\\textnormal{ oraz } x \\in \\left(- \\frac{1}{2},\\frac{1}{2}  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }\\frac{1}{2} < x \\vee x < - \\frac{3}{2} \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( -\\infty, - \\frac{3}{2}  \\right)\\textnormal{ oraz } x \\in \\left(\\frac{1}{2}, \\infty  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace - \\frac{3}{2}, \\  \\frac{1}{2}\\right\\rbrace \\implies f_{\\min}\\left(- \\frac{3}{2}\\right) = 6 \\textnormal{ oraz } f_{\\max}\\left(\\frac{1}{2}\\right) = -2$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 5 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{2 x - 1}{- 2 x^{2} + 2 x - 1} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }1 < x \\vee x < 0 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left( -\\infty, 0  \\right)\\textnormal{ oraz } x \\in \\left(1, \\infty    \\right)$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }0 < x \\wedge x < 1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( 0, 1  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace 0, \\  1\\right\\rbrace \\implies f_{\\max}\\left(0\\right) = 1 \\textnormal{ oraz } f_{\\min}\\left(1\\right) = -1$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 6 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{4 - 2 x}{- x^{2} - x + 2} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace -2, \\  1\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x > 0 \\wedge x < 4 \\wedge x \\neq 1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(0,1  \\right)\\textnormal{ oraz } x \\in \\left(1,4  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }\\left(-2 < x \\wedge x < 0\\right) \\vee 4 < x \\vee x < -2 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( -\\infty, -2  \\right)\\textnormal{ oraz } x \\in \\left(-2, 0  \\right) \\textnormal{ oraz } x \\in \\left(4, \\infty  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace 0, \\  4\\right\\rbrace \\implies f_{\\min}\\left(0\\right) = 2 \\textnormal{ oraz } f_{\\max}\\left(4\\right) = \\frac{2}{9}$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 7 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{x^{3}}{- x - 2} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace-2\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x < -3 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(-\\infty,-3\\right)  $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }\\left(-3 < x \\wedge x < -2\\right) \\vee \\left(-2 < x \\wedge x < 0\\right) \\vee 0 < x \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left(-3,-2\\right) \\textnormal{ oraz } x \\in  \\left(-2,\\infty\\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace -3, \\  0\\right\\rbrace \\implies f_{\\max}\\left(-3\\right) = -27$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 8 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=- \\frac{2 x^{3}}{3 x + 1} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace- \\frac{1}{3}\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x < - \\frac{1}{2} \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(-\\infty,- \\frac{1}{2}\\right)  $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }\\left(- \\frac{1}{2} < x \\wedge x < - \\frac{1}{3}\\right) \\vee \\left(- \\frac{1}{3} < x \\wedge x < 0\\right) \\vee 0 < x \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left(- \\frac{1}{2},- \\frac{1}{3}\\right) \\textnormal{ oraz } x \\in  \\left(- \\frac{1}{3},\\infty\\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace - \\frac{1}{2}, \\  0\\right\\rbrace \\implies f_{\\max}\\left(- \\frac{1}{2}\\right) = - \\frac{1}{2}$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 9 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{x^{2} + 2 x + 1}{3 - 2 x} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace\\frac{3}{2}\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x > -1 \\wedge x < 4 \\wedge x \\neq \\frac{3}{2} \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(-1,\\frac{3}{2}  \\right)\\textnormal{ oraz } x \\in \\left(\\frac{3}{2},4  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }4 < x \\vee x < -1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( -\\infty, -1  \\right)\\textnormal{ oraz } x \\in \\left(4, \\infty  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace -1, \\  4\\right\\rbrace \\implies f_{\\min}\\left(-1\\right) = 0 \\textnormal{ oraz } f_{\\max}\\left(4\\right) = -5$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n",
      "\u001B[34m** Zadanie 10 **********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "Zbadać monotoniczność i ekstrema funkcji"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": "$$\tf(x)=\\frac{- x - 2}{- x^{2} + x + 2} $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\u001B[34m*\u001B[0m\n",
      "\u001B[32m** Rozwiązanie ********************************************************************************\u001B[0m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$D_f\\colon \\mathbb{R}\\setminus \\left\\lbrace -1, \\  2\\right\\rbrace$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) > 0 \\textnormal{ dla: }x > -4 \\wedge x < 0 \\wedge x \\neq -1 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest rosnąca dla } x \\in \\left(-4,-1  \\right)\\textnormal{ oraz } x \\in \\left(-1,0  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) < 0 \\textnormal{ dla: }\\left(0 < x \\wedge x < 2\\right) \\vee 2 < x \\vee x < -4 \\implies$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$\\implies f(x) \\textnormal{ jest malejąca dla } x \\in \\left( -\\infty, -4  \\right)\\textnormal{ oraz } x \\in \\left(0, 2  \\right) \\textnormal{ oraz } x \\in \\left(2, \\infty  \\right) $$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ],
      "text/markdown": "$$f'(x) = 0 \\textnormal{ dla } x \\in \\left\\lbrace -4, \\  0\\right\\rbrace \\implies f_{\\min}\\left(-4\\right) = - \\frac{1}{9} \\textnormal{ oraz } f_{\\max}\\left(0\\right) = -1$$"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\u001B[32m*\u001B[0m\n"
     ]
    }
   ],
   "execution_count": 8
  },
  {
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-01-28T11:14:58.175241Z",
     "start_time": "2025-01-28T11:14:58.169383Z"
    }
   },
   "cell_type": "code",
   "source": "",
   "id": "4312dfd0086210b2",
   "outputs": [],
   "execution_count": null
  }
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