{"id":2732,"date":"2023-03-03T09:30:57","date_gmt":"2023-03-03T08:30:57","guid":{"rendered":"http:\/\/matma.com.pl\/?p=2732"},"modified":"2023-04-18T17:30:21","modified_gmt":"2023-04-18T15:30:21","slug":"zadanie-27-0-6-zbior-rozszerzenie","status":"publish","type":"post","link":"http:\/\/matma.com.pl\/?p=2732","title":{"rendered":"zadanie 27 (0-6) zbi\u00f3r rozszerzenie"},"content":{"rendered":"<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">Ustalmy boki naszego tr\u00f3jk\u0105ta bazuj\u0105c na danych z zadania i opiszmy na nim okr\u0105g.<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;BC&amp;space;\\right&amp;space;|=x\" alt=\"\\left | BC \\right |=x\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|AB&amp;space;\\right&amp;space;|=2x\" alt=\"\\left |AB \\right |=2x\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?Obw_{\\bigtriangleup&amp;space;ABC}=\\left&amp;space;|&amp;space;AB&amp;space;\\right&amp;space;|+\\left&amp;space;|&amp;space;BC&amp;space;\\right&amp;space;|+\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|\" alt=\"Obw_{\\bigtriangleup ABC}=\\left | AB \\right |+\\left | BC \\right |+\\left | AC \\right |\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?3R=2x+x+\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|\" alt=\"3R=2x+x+\\left | AC \\right |\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|=3R-3x\" alt=\"\\left | AC \\right |=3R-3x\" align=\"absmiddle\" \/><\/td>\n<td style=\"width: 50%;\"><img fetchpriority=\"high\" decoding=\"async\" class=\" wp-image-2734 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-2023-02-28-090717-294x300.jpg\" alt=\"\" width=\"332\" height=\"338\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-2023-02-28-090717-294x300.jpg 294w, http:\/\/matma.com.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-2023-02-28-090717.jpg 372w\" sizes=\"(max-width: 332px) 100vw, 332px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Z nier\u00f3wno\u015bci tr\u00f3jk\u0105ta mamy:<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;AB&amp;space;\\right&amp;space;|+\\left&amp;space;|BC&amp;space;\\right&amp;space;|&gt;\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|\" alt=\"\\left | AB \\right |+\\left |BC \\right |&gt;\\left | AC \\right |\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?2x+x&gt;3R-3x\" alt=\"2x+x&gt;3R-3x\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?6x&gt;3R\\;&amp;space;\\;&amp;space;\/:6\" alt=\"6x&gt;3R\\; \\; \/:6\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x&gt;&amp;space;\\frac{1}{2}R\" alt=\"x&gt; \\frac{1}{2}R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;AB&amp;space;\\right&amp;space;|+\\left&amp;space;|AC&amp;space;\\right&amp;space;|&gt;\\left&amp;space;|&amp;space;BC&amp;space;\\right&amp;space;|\" alt=\"\\left | AB \\right |+\\left |AC \\right |&gt;\\left | BC \\right |\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?2x+3R-3x&gt;x\" alt=\"2x+3R-3x&gt;x\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-2x&gt;-3R\\;&amp;space;\\;&amp;space;\/:\\left&amp;space;(&amp;space;-2&amp;space;\\right&amp;space;)\" alt=\"-2x&gt;-3R\\; \\; \/:\\left ( -2 \\right )\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x&lt;&amp;space;\\frac{3}{2}R\" alt=\"x&lt; \\frac{3}{2}R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|+\\left&amp;space;|BC&amp;space;\\right&amp;space;|&gt;\\left&amp;space;|&amp;space;AB&amp;space;\\right&amp;space;|\" alt=\"\\left | AC \\right |+\\left |BC \\right |&gt;\\left | AB \\right |\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?3R-3x+x&gt;&amp;space;2x\" alt=\"3R-3x+x&gt; 2x\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?-4x&gt;&amp;space;-3R\\;&amp;space;\\;&amp;space;\/:\\left&amp;space;(&amp;space;-4&amp;space;\\right&amp;space;)\" alt=\"-4x&gt; -3R\\; \\; \/:\\left ( -4 \\right )\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x&lt;&amp;space;\\frac{3}{4}R\" alt=\"x&lt; \\frac{3}{4}R\" align=\"absmiddle\" \/><\/p>\n<p>Zatem <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{1}{2}R,\\frac{3}{4}R&amp;space;\\right&amp;space;)\" alt=\"x\\in \\left ( \\frac{1}{2}R,\\frac{3}{4}R \\right )\" align=\"absmiddle\" \/><\/p>\n<p>Skorzystamy z wzoru na pole tr\u00f3jk\u0105ta uwzgl\u0119dniaj\u0105cy promie\u0144 okr\u0119gu opisanego na tr\u00f3jk\u0105cie:<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P_{\\bigtriangleup&amp;space;ABC}=\\frac{\\left&amp;space;|&amp;space;AB&amp;space;\\right&amp;space;|\\cdot&amp;space;\\left&amp;space;|&amp;space;BC&amp;space;\\right&amp;space;|\\cdot&amp;space;\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|}{4R}\" alt=\"P_{\\bigtriangleup ABC}=\\frac{\\left | AB \\right |\\cdot \\left | BC \\right |\\cdot \\left | AC \\right |}{4R}\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P_{\\bigtriangleup&amp;space;ABC}=\\frac{2x\\cdot&amp;space;x\\cdot&amp;space;\\left&amp;space;(&amp;space;3R-3x&amp;space;\\right&amp;space;)}{4R}=\\frac{6x^2R-6x^3}{4}=-\\frac{3}{2}x^3+\\frac{3}{2}x^2R\" alt=\"P_{\\bigtriangleup ABC}=\\frac{2x\\cdot x\\cdot \\left ( 3R-3x \\right )}{4R}=\\frac{6x^2R-6x^3}{4}=-\\frac{3}{2}x^3+\\frac{3}{2}x^2R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P(x)=-\\frac{3}{2}x^3+\\frac{3}{2}x^2R\" alt=\"P(x)=-\\frac{3}{2}x^3+\\frac{3}{2}x^2R\" align=\"absmiddle\" \/> dla <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{1}{2}R,\\frac{3}{4}R&amp;space;\\right&amp;space;)\" alt=\"x\\in \\left ( \\frac{1}{2}R,\\frac{3}{4}R \\right )\" align=\"absmiddle\" \/><\/p>\n<p>Obliczymy nast\u0119pnie pochodn\u0105 <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P'(x)\" alt=\"P'(x)\" align=\"absmiddle\" \/> oraz jej miejsca zerowe.<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P'(x)=-\\frac{3}{2}\\cdot&amp;space;3x^2+\\frac{3}{2}\\cdot&amp;space;2xR\" alt=\"P'(x)=-\\frac{3}{2}\\cdot 3x^2+\\frac{3}{2}\\cdot 2xR\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P'(x)=-\\frac{9}{2}x^2+3xR\" alt=\"P'(x)=-\\frac{9}{2}x^2+3xR\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P'(x)=0\\;&amp;space;\\;&amp;space;\\Leftrightarrow&amp;space;\\;&amp;space;\\;&amp;space;-\\frac{9}{2}x^2+3xR=0\" alt=\"P'(x)=0\\; \\; \\Leftrightarrow \\; \\; -\\frac{9}{2}x^2+3xR=0\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\left&amp;space;(&amp;space;-\\frac{9}{2}x+3R&amp;space;\\right&amp;space;)=0\" alt=\"x\\left ( -\\frac{9}{2}x+3R \\right )=0\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=0\\;&amp;space;\\;&amp;space;\\vee&amp;space;\\;&amp;space;\\;&amp;space;-\\frac{9}{2}x+3R=0\" alt=\"x=0\\; \\; \\vee \\; \\; -\\frac{9}{2}x+3R=0\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=0\\;&amp;space;\\;&amp;space;\\vee&amp;space;\\;&amp;space;\\;&amp;space;-\\frac{9}{2}x=-3R\\;&amp;space;\\;&amp;space;\/\\cdot&amp;space;\\left&amp;space;(-\\frac{2}{9}&amp;space;\\right&amp;space;)\" alt=\"x=0\\; \\; \\vee \\; \\; -\\frac{9}{2}x=-3R\\; \\; \/\\cdot \\left (-\\frac{2}{9} \\right )\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=0\\;&amp;space;\\;&amp;space;\\vee&amp;space;\\;&amp;space;\\;&amp;space;x=\\frac{2}{3}R\" alt=\"x=0\\; \\; \\vee \\; \\; x=\\frac{2}{3}R\" align=\"absmiddle\" \/><\/p>\n<p>Uwzgl\u0119dniaj\u0105c\u00a0<img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{1}{2}R,\\frac{3}{4}R&amp;space;\\right&amp;space;)\" alt=\"x\\in \\left ( \\frac{1}{2}R,\\frac{3}{4}R \\right )\" align=\"absmiddle\" \/> mamy <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x_{0}=\\frac{2}{3}R\" alt=\"x_{0}=\\frac{2}{3}R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P\\left&amp;space;(\\frac{2}{3}&amp;space;R&amp;space;\\right&amp;space;)=&amp;space;-\\frac{3}{2}\\cdot&amp;space;\\left&amp;space;(&amp;space;\\frac{2}{3}R&amp;space;\\right&amp;space;)^3+\\frac{3}{2}\\left&amp;space;(&amp;space;\\frac{2}{3}R&amp;space;\\right&amp;space;)^2\\cdot&amp;space;R=-\\frac{3}{2}\\cdot&amp;space;\\frac{8}{27}R^3+\\frac{3}{2}\\cdot&amp;space;\\frac{4}{9}R^2\\cdot&amp;space;R\" alt=\"P\\left (\\frac{2}{3} R \\right )= -\\frac{3}{2}\\cdot \\left ( \\frac{2}{3}R \\right )^3+\\frac{3}{2}\\left ( \\frac{2}{3}R \\right )^2\\cdot R=-\\frac{3}{2}\\cdot \\frac{8}{27}R^3+\\frac{3}{2}\\cdot \\frac{4}{9}R^2\\cdot R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P\\left&amp;space;(\\frac{2}{3}&amp;space;R&amp;space;\\right&amp;space;)=&amp;space;-\\frac{4}{9}R^3+\\frac{2}{3}R^3=\\frac{2}{9}R^2\" alt=\"P\\left (\\frac{2}{3} R \\right )= -\\frac{4}{9}R^3+\\frac{2}{3}R^3=\\frac{2}{9}R^2\" align=\"absmiddle\" \/><\/p>\n<p>Zbadajmy przebieg zmienno\u015bci funkcji<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{1}{2}R,\\frac{3}{4}R&amp;space;\\right&amp;space;)\" alt=\"x\\in \\left ( \\frac{1}{2}R,\\frac{3}{4}R \\right )\" align=\"absmiddle\" \/><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P'(x)&gt;&amp;space;0\\;&amp;space;\\;&amp;space;\\Leftrightarrow&amp;space;\\;&amp;space;\\;&amp;space;x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{1}{2}R&amp;space;,\\frac{2}{3}R\\right&amp;space;)\" alt=\"P'(x)&gt; 0\\; \\; \\Leftrightarrow \\; \\; x\\in \\left ( \\frac{1}{2}R ,\\frac{2}{3}R\\right )\" align=\"absmiddle\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P'(x)&lt;&amp;space;0\\;&amp;space;\\;&amp;space;\\Leftrightarrow&amp;space;\\;&amp;space;\\;&amp;space;x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{2}{3}R&amp;space;,\\frac{3}{4}R&amp;space;\\right&amp;space;)\" alt=\"P'(x)&lt; 0\\; \\; \\Leftrightarrow \\; \\; x\\in \\left ( \\frac{2}{3}R ,\\frac{3}{4}R \\right )\" align=\"absmiddle\" \/><\/p>\n<p>&nbsp;<\/td>\n<td style=\"width: 50%;\"><img decoding=\"async\" class=\"wp-image-2742 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-2023-02-28-110435-300x144.jpg\" alt=\"\" width=\"363\" height=\"174\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-2023-02-28-110435-300x144.jpg 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2023\/02\/Zrzut-ekranu-2023-02-28-110435.jpg 568w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Podsumowuj\u0105c <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P(x)\" alt=\"P(x)\" align=\"absmiddle\" \/> jest rosn\u0105ca w przedziale <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\in&amp;space;\\left&amp;space;(&amp;space;\\frac{1}{2}R&amp;space;,\\frac{2}{3}R&amp;space;\\right&amp;space;]\" alt=\"x\\in \\left ( \\frac{1}{2}R ,\\frac{2}{3}R \\right ]\" align=\"absmiddle\" \/> a malej\u0105ca w przedziale <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x\\in\\left&amp;space;[&amp;space;\\frac{2}{3}R&amp;space;,\\frac{3}{4}R&amp;space;\\right&amp;space;)\" alt=\"x\\in\\left [ \\frac{2}{3}R ,\\frac{3}{4}R \\right )\" align=\"absmiddle\" \/>. Oznacza to, \u017ce <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=\\frac{2}{3}R\" alt=\"x=\\frac{2}{3}R\" align=\"absmiddle\" \/> funkcja osi\u0105ga maksimum.<\/p>\n<p>Najwi\u0119ksze pole tr\u00f3jk\u0105ta jest r\u00f3wne <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\frac{2}{9}R^2\" alt=\"\\frac{2}{9}R^2\" align=\"absmiddle\" \/> dla <img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?x=\\frac{2}{3}R\" alt=\"x=\\frac{2}{3}R\" align=\"absmiddle\" \/>.<\/p>\n<p>D\u0142ugo\u015bci bok\u00f3w naszego tr\u00f3jk\u0105ta s\u0105 r\u00f3wne:<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;AB&amp;space;\\right&amp;space;|=2\\cdot&amp;space;\\frac{2}{3}R=\\frac{4}{3}R\" alt=\"\\left | AB \\right |=2\\cdot \\frac{2}{3}R=\\frac{4}{3}R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;BC&amp;space;\\right&amp;space;|=&amp;space;\\frac{2}{3}R\" alt=\"\\left | BC \\right |= \\frac{2}{3}R\" align=\"absmiddle\" \/><\/p>\n<p><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?\\left&amp;space;|&amp;space;AC&amp;space;\\right&amp;space;|=3R-3\\cdot&amp;space;\\frac{2}{3}R=3R-2R=R\" alt=\"\\left | AC \\right |=3R-3\\cdot \\frac{2}{3}R=3R-2R=R\" align=\"absmiddle\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ustalmy boki naszego tr\u00f3jk\u0105ta bazuj\u0105c na danych z zadania i opiszmy na nim okr\u0105g. Z nier\u00f3wno\u015bci tr\u00f3jk\u0105ta mamy: Zatem Skorzystamy z wzoru na pole tr\u00f3jk\u0105ta uwzgl\u0119dniaj\u0105cy promie\u0144 okr\u0119gu opisanego na tr\u00f3jk\u0105cie: dla Obliczymy nast\u0119pnie pochodn\u0105 oraz jej miejsca zerowe. Uwzgl\u0119dniaj\u0105c\u00a0 mamy Zbadajmy przebieg zmienno\u015bci funkcji &nbsp; &nbsp; Podsumowuj\u0105c jest rosn\u0105ca w przedziale a malej\u0105ca w [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2732","post","type-post","status-publish","format-standard","hentry","category-najnowsze-pliki"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>zadanie 27 (0-6) zbi\u00f3r rozszerzenie - Matma.com.pl<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/matma.com.pl\/?p=2732\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"zadanie 27 (0-6) zbi\u00f3r rozszerzenie - Matma.com.pl\" \/>\n<meta property=\"og:description\" content=\"Ustalmy boki naszego tr\u00f3jk\u0105ta bazuj\u0105c na danych z zadania i opiszmy na nim okr\u0105g. Z nier\u00f3wno\u015bci tr\u00f3jk\u0105ta mamy: Zatem Skorzystamy z wzoru na pole tr\u00f3jk\u0105ta uwzgl\u0119dniaj\u0105cy promie\u0144 okr\u0119gu opisanego na tr\u00f3jk\u0105cie: dla Obliczymy nast\u0119pnie pochodn\u0105 oraz jej miejsca zerowe. 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