{"id":2952,"date":"2024-02-28T12:41:05","date_gmt":"2024-02-28T12:41:05","guid":{"rendered":"https:\/\/bezwarunku.pl\/?page_id=2952"},"modified":"2024-05-15T11:12:20","modified_gmt":"2024-05-15T10:12:20","slug":"punkt-przegiecia-wypuklosc-i-wkleslosc-funkcji","status":"publish","type":"page","link":"https:\/\/bezwarunku.pl\/index.php\/punkt-przegiecia-wypuklosc-i-wkleslosc-funkcji\/","title":{"rendered":"punkt przegi\u0119cia, wypuk\u0142o\u015b\u0107 i wkl\u0119s\u0142o\u015b\u0107 funkcji"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"2952\" class=\"elementor elementor-2952\" data-elementor-post-type=\"page\">\n\t\t\t\t<header class=\"elementor-element elementor-element-fdb1ceb e-con-full e-flex e-con e-parent\" data-id=\"fdb1ceb\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t<div class=\"elementor-element elementor-element-725a3bd e-con-full e-flex e-con e-child\" 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class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/bezwarunku.pl\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">zarejestruj si\u0119<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/header>\n\t\t<div class=\"elementor-element elementor-element-8a8b8b0 e-con-full e-flex e-con e-parent\" data-id=\"8a8b8b0\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0e27ee2 elementor-widget elementor-widget-heading\" data-id=\"0e27ee2\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/bezwarunku.pl\/index.php\/analiza-matematyczna\/?preview_id=200&#038;preview_nonce=87c719741f&#038;preview=true\">POCHODNE<\/a><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b937c07 animated-slow elementor-invisible elementor-widget elementor-widget-heading\" data-id=\"b937c07\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;slideInUp&quot;}\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">punkt przegi\u0119cia, wypuk\u0142o\u015b\u0107 i wkl\u0119s\u0142o\u015b\u0107 funkcji <\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-4d099a7 e-flex e-con-boxed e-con e-parent\" data-id=\"4d099a7\" data-element_type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-166ccb9 e-con-full e-flex e-con e-child\" data-id=\"166ccb9\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f3bf2f5 elementor-widget elementor-widget-heading\" data-id=\"f3bf2f5\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">PUNKT PRZEGI\u0118CIA<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-65343c1 elementor-widget elementor-widget-image\" data-id=\"65343c1\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"299\" height=\"295\" src=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/punkt-przegiecia.png\" class=\"attachment-large size-large wp-image-3036\" alt=\"\" srcset=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/punkt-przegiecia.png 299w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/punkt-przegiecia-100x100.png 100w\" sizes=\"(max-width: 299px) 100vw, 299px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d7c4256 elementor-widget elementor-widget-text-editor\" data-id=\"d7c4256\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u00a0Punkt przegi\u0119cia to miejsce, w kt\u00f3rym krzywa przestaje si\u0119 wygina\u0107 w jednym kierunku i zaczyna wygina\u0107 si\u0119 w drugim. Czyli punkt gdzie funkcja zmienia si\u0119 z wkl\u0119s\u0142ej na wypuk\u0142\u0105 lub odwrotnie.<\/p><p>Jak obliczy\u0107?:<\/p><ol><li>Obliczasz I i II pochodn\u0105 funkcji<\/li><li>Przyr\u00f3wnujesz do 0<\/li><li>Sprawdzasz czy wynik nale\u017cy do dziedziny<\/li><li>sprawdzasz znaki po jednej i drugiej stronie funkcji<\/li><li>jak z dw\u00f3ch stron s\u0105 r\u00f3\u017cne znaki, mamy punkt przegi\u0119cia.<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-4d8725f e-con-full e-flex e-con e-child\" data-id=\"4d8725f\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d99a2e8 elementor-widget elementor-widget-heading\" data-id=\"d99a2e8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">FUNKCJA WYPUK\u0141A<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-268db29 elementor-widget elementor-widget-image\" data-id=\"268db29\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"295\" height=\"287\" src=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/wypukla.png\" class=\"attachment-large size-large wp-image-3037\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-958dc1a elementor-widget elementor-widget-text-editor\" data-id=\"958dc1a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Jak obliczy\u0107?:<\/p><ol><li>Obliczasz I i II pochodn\u0105 funkcji<\/li><li>Przyr\u00f3wnujesz do nier\u00f3wno\u015bci &gt;<\/li><li>Rozwi\u0105zujesz nier\u00f3wno\u015b\u0107<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-e7d5d5d e-con-full e-flex e-con e-child\" data-id=\"e7d5d5d\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-18f650d elementor-widget elementor-widget-heading\" data-id=\"18f650d\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">FUNKCJA WKL\u0118S\u0141A<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e162a83 elementor-widget elementor-widget-image\" data-id=\"e162a83\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"285\" height=\"289\" src=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/wklesla.png\" class=\"attachment-large size-large wp-image-3038\" alt=\"\" srcset=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/wklesla.png 285w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/wklesla-100x100.png 100w\" sizes=\"(max-width: 285px) 100vw, 285px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-857078f elementor-widget elementor-widget-text-editor\" data-id=\"857078f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Jak obliczy\u0107?:<\/p><ol><li>Obliczasz I i II pochodn\u0105 funkcji<\/li><li>Przyr\u00f3wnujesz do nier\u00f3wno\u015bci &lt; 0<\/li><li>Rozwi\u0105zujesz nier\u00f3wno\u015b\u0107<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-6afe47f e-flex e-con-boxed e-con e-parent\" data-id=\"6afe47f\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-0b52d6e e-con-full e-flex e-con e-child\" data-id=\"0b52d6e\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-16fc90e elementor-widget elementor-widget-toggle\" data-id=\"16fc90e\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4b7a48a elementor-widget elementor-widget-heading\" data-id=\"4b7a48a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">1. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=x^3-9x^2+1$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-6554ea7 e-con-full e-flex e-con e-child\" data-id=\"6554ea7\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6c15008 elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"6c15008\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-388e346 elementor-widget elementor-widget-heading\" data-id=\"388e346\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">2. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=x^4-x^5$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-75e5de4 e-con-full e-flex e-con e-child\" data-id=\"75e5de4\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b799cff elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"b799cff\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3ae563f elementor-widget elementor-widget-heading\" data-id=\"3ae563f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">3. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=x+\\frac1x$$<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-7fd0fbe e-con-full e-flex e-con e-child\" data-id=\"7fd0fbe\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-99f403b elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"99f403b\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4970f95 elementor-widget elementor-widget-heading\" data-id=\"4970f95\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">4. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=\\frac{3x^2}{1-x}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-766aab4 e-con-full e-flex e-con e-child\" data-id=\"766aab4\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d568b3d elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"d568b3d\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-76eed3a elementor-widget elementor-widget-heading\" data-id=\"76eed3a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">5. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=\\frac x{1-x^2}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-8a86127 e-flex e-con-boxed e-con e-parent\" data-id=\"8a86127\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-17846bd e-con-full e-flex e-con e-child\" data-id=\"17846bd\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f929f6c elementor-widget elementor-widget-toggle\" data-id=\"f929f6c\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h4 id=\"elementor-tab-title-2611\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2611\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><svg class=\"e-font-icon-svg e-fas-caret-right\" viewBox=\"0 0 192 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M0 384.662V127.338c0-17.818 21.543-26.741 34.142-14.142l128.662 128.662c7.81 7.81 7.81 20.474 0 28.284L34.142 398.804C21.543 411.404 0 402.48 0 384.662z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><svg class=\"elementor-toggle-icon-opened e-font-icon-svg e-fas-caret-up\" viewBox=\"0 0 320 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M288.662 352H31.338c-17.818 0-26.741-21.543-14.142-34.142l128.662-128.662c7.81-7.81 20.474-7.81 28.284 0l128.662 128.662c12.6 12.599 3.676 34.142-14.142 34.142z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">rozwi\u0105zanie<\/a>\n\t\t\t\t\t<\/h4>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2611\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2611\"><p>\\[ f\\left(x\\right)=\\frac1{x^3}+\\frac1{x^2}\\]<br \/>Dziedzina funkcji: \\( D=\\mathbb{R}\\backslash\\left\\{0\\right\\}\\)<\/p><p>Pierwsza pochodna funkcji:<br \/>$$f&#8217;\\left(x\\right)=-\\frac{3+2x}{x^4}$$<\/p><p>Druga pochodna funkcji:<\/p><p>$$f&#8221;\\left(x\\right)=\\frac{12+6x}{x^5}$$<\/p><p>Wyznaczmy punkty przegi\u0119cia:<\/p><p>$$f&#8221;\\left(x\\right)=0\\\\\\frac{12+6x}{x^5}=0\\\\12+6x=0\\\\x=-2$$<\/p><p>czyli punkt x=-2 jest podejrzany o bycie punktem przegi\u0119cia. Sprawdzamy znaki po lewej i prawej stronie funkcji.\u00a0<\/p><p>Zacznijmy od lewej strony. Bierzesz dowolny punkt na lewo od -2, np. -3.<\/p><p>Podstawiamy ten punkt pod funkcj\u0119 na drug\u0105 pochodn\u0105:$$f&#8221;\\left(-3\\right)=\\frac{12+6\\cdot\\left(-3\\right)}{\\left(-3\\right)^5}=\\frac2{81}$$<\/p><p>Wynik jest dodatni<\/p><p>Teraz dowolny punkt na prawo w przedziale (-2, 0) np. -1. Podstawiamy ten punkt pod funkcj\u0119 na drug\u0105 pochodn\u0105: $$f&#8221;\\left(-1\\right)=\\frac{12+6\\cdot\\left(-1\\right)}{\\left(-1\\right)^5}=-6$$<\/p><p>Wynik jest ujemny<\/p><p>Otrzymali\u015bmy z obydwu stron r\u00f3\u017cne znaki, czyli x=-2 jest naszym punktem przegi\u0119cia.<\/p><p>Warto\u015b\u0107 dla tego punktu: $$f\\left(-2\\right)=\\frac1{\\left(-2\\right)^3}+\\frac1{\\left(-2\\right)^2}=\\frac18$$<\/p><p>Wyznaczmy przedzia\u0142 wkl\u0119s\u0142o\u015bci:<\/p><p>$$f&#8221;\\left(x\\right)&lt;0\\\\\\frac{12+6x}{x^5}&lt;0\\\\x\\in\\left(-2,0\\right)$$<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-3015 aligncenter\" src=\"http:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/zzzzzz-300x219.png\" alt=\"\" width=\"300\" height=\"219\" srcset=\"https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/zzzzzz-300x219.png 300w, https:\/\/bezwarunku.pl\/wp-content\/uploads\/2024\/02\/zzzzzz.png 405w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Przedzia\u0142 wypuk\u0142o\u015bci:<\/p><p>$$f&#8221;\\left(x\\right)&gt;0\\\\\\frac{12+6x}{x^5}&gt;0\\\\x\\in\\left(-\\infty,-2\\right)\\vee\\left(0,+\\infty\\right)$$<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7de16f2 elementor-widget elementor-widget-heading\" data-id=\"7de16f2\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">6. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=\\frac1{x^3}+\\frac1{x^2}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-5dc797b e-con-full e-flex e-con e-child\" data-id=\"5dc797b\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-bde1f73 elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"bde1f73\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b31a50a elementor-widget elementor-widget-heading\" data-id=\"b31a50a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">7. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=3x-\\sqrt{x-3}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-22af865 e-con-full e-flex e-con e-child\" data-id=\"22af865\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-370a434 elementor-widget__width-initial elementor-widget elementor-widget-toggle\" data-id=\"370a434\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pmpro\"><div class=\"pmpro_card pmpro_content_message\"><div class=\"pmpro_card_content\">nie czekaj d\u0142u\u017cej, wykup pakiet i sprawd\u017a rozwi\u0105zania.<br \/><a href=\"https:\/\/bezwarunku.pl\/index.php\/wykup-dostep\/\">Przed\u0142u\u017c pakiet!<\/a><\/div><div class=\"pmpro_card_actions pmpro_font-medium\">Already a member? <a href=\"https:\/\/bezwarunku.pl\/index.php\/membership-login\/?redirect_to=https%3A%2F%2Fbezwarunku.pl%2Findex.php%2Fpunkt-przegiecia-wypuklosc-i-wkleslosc-funkcji%2F\">Log in here<\/a><\/div><\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-609acee elementor-widget elementor-widget-heading\" data-id=\"609acee\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">8. Wyznacz punkty przegi\u0119cia oraz przedzia\u0142y wkl\u0119s\u0142o\u015bci i wypuk\u0142o\u015bci funkcji: $$f\\left(x\\right)=xe^{-x}$$<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>zarejestruj si\u0119 POCHODNE punkt przegi\u0119cia, wypuk\u0142o\u015b\u0107 i wkl\u0119s\u0142o\u015b\u0107 funkcji PUNKT PRZEGI\u0118CIA \u00a0Punkt przegi\u0119cia to miejsce, w kt\u00f3rym krzywa przestaje si\u0119 wygina\u0107 w jednym kierunku i zaczyna wygina\u0107 si\u0119 w drugim. Czyli punkt gdzie funkcja zmienia si\u0119 z wkl\u0119s\u0142ej na wypuk\u0142\u0105 lub odwrotnie. Jak obliczy\u0107?: Obliczasz I i II pochodn\u0105 funkcji Przyr\u00f3wnujesz do 0 Sprawdzasz czy [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/landing.php","meta":{"pmpro_default_level":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":""},"class_list":["post-2952","page","type-page","status-publish","hentry","pmpro-has-access","entry","owp-thumbs-layout-horizontal","owp-btn-normal","owp-tabs-layout-horizontal","has-no-thumbnails","has-product-nav"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages\/2952","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/comments?post=2952"}],"version-history":[{"count":73,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages\/2952\/revisions"}],"predecessor-version":[{"id":4576,"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/pages\/2952\/revisions\/4576"}],"wp:attachment":[{"href":"https:\/\/bezwarunku.pl\/index.php\/wp-json\/wp\/v2\/media?parent=2952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}