{"id":41,"date":"2023-09-17T10:59:38","date_gmt":"2023-09-17T10:59:38","guid":{"rendered":"https:\/\/mathority.org\/ru\/%d0%b4%d0%b8%d1%84%d1%84%d0%b5%d1%80%d0%b5%d0%bd%d1%86%d0%b8%d1%80%d1%83%d0%b5%d0%bc%d0%be%d1%81%d1%82%d1%8c-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8\/"},"modified":"2023-09-17T10:59:38","modified_gmt":"2023-09-17T10:59:38","slug":"%d0%b4%d0%b8%d1%84%d1%84%d0%b5%d1%80%d0%b5%d0%bd%d1%86%d0%b8%d1%80%d1%83%d0%b5%d0%bc%d0%be%d1%81%d1%82%d1%8c-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8","status":"publish","type":"post","link":"https:\/\/mathority.org\/ru\/%d0%b4%d0%b8%d1%84%d1%84%d0%b5%d1%80%d0%b5%d0%bd%d1%86%d0%b8%d1%80%d1%83%d0%b5%d0%bc%d0%be%d1%81%d1%82%d1%8c-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8\/","title":{"rendered":"\u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438"},"content":{"rendered":"<p>\u0412 \u044d\u0442\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u0432\u044b \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0438\u0437\u0443\u0447\u0430\u0442\u044c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0442\u043e \u0435\u0441\u0442\u044c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0438\u043b\u0438 \u043d\u0435\u0442. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c \u0441\u0432\u044f\u0437\u044c \u043c\u0435\u0436\u0434\u0443 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c\u044e \u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u0438\u0437\u0443\u0447\u0438\u043c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043a\u0443\u0441\u043e\u0447\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-y-continuidad-de-una-funcion\"><\/span> \u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>\u041d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c<\/strong> \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u043e\u0447\u043a\u0435 \u0441\u0432\u044f\u0437\u0430\u043d\u044b \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<ul>\n<li> \u0415\u0441\u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u0442\u043e\u0447\u043a\u0435, \u0442\u043e \u043e\u043d\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435.<\/li>\n<li> \u0415\u0441\u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439 \u0432 \u043a\u0430\u043a\u043e\u0439-\u0442\u043e \u0442\u043e\u0447\u043a\u0435, \u043e\u043d\u0430 \u0442\u0430\u043a\u0436\u0435 \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435.<\/li>\n<\/ul>\n<p> \u041e\u0434\u043d\u0430\u043a\u043e \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0435 \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0435 \u044d\u0442\u043e\u0439 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u043d\u0435\u0432\u0435\u0440\u043d\u043e: \u0442\u043e, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u043a\u0430\u043a\u043e\u0439-\u0442\u043e \u0442\u043e\u0447\u043a\u0435, \u043d\u0435 \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u043e\u043d\u0430 \u0432\u0441\u0435\u0433\u0434\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435.<\/p>\n<p> \u0412\u044b \u0442\u0430\u043a\u0436\u0435 \u043c\u043e\u0436\u0435\u0442\u0435 \u0443\u0432\u0438\u0434\u0435\u0442\u044c, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0435, \u043f\u043e \u0435\u0435 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u043c\u0443 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044e:<\/p>\n<ul>\n<li> \u0415\u0441\u043b\u0438 \u044d\u0442\u043e <strong>\u0433\u043b\u0430\u0434\u043a\u0430\u044f \u0442\u043e\u0447\u043a\u0430,<\/strong> \u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430.<\/li>\n<li> \u0415\u0441\u043b\u0438 \u044d\u0442\u043e <strong>\u0443\u0433\u043b\u043e\u0432\u0430\u044f \u0442\u043e\u0447\u043a\u0430,<\/strong> \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430, \u043d\u043e \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430. <\/li>\n<\/ul>\n<div class=\"wp-block-columns is-layout-flex wp-container-15\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-pour-representer-une-fonction-quadratique-incomplete.webp\" alt=\"\" class=\"wp-image-140\" width=\"250\" height=\"279\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <span style=\"color:#1976d2;\"><strong>\u0422\u043e\u0447\u043a\u0430 \u0441\u0433\u043b\u0430\u0436\u0438\u0432\u0430\u043d\u0438\u044f<\/strong><\/span> \u043f\u0440\u0438 x=0:<br \/> \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430\u044f \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435. <\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/comment-representer-graphiquement-une-fonction-avec-valeur-absolue.webp\" alt=\"\" class=\"wp-image-230\" width=\"286\" height=\"305\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <span style=\"color:#1976d2;\"><strong>\u0423\u0433\u043b\u043e\u0432\u0430\u044f \u0442\u043e\u0447\u043a\u0430<\/strong><\/span> \u043f\u0440\u0438 x=2:<br \/> \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430, \u043d\u043e \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435. <\/p>\n<\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-de-una-funcion-a-trozos\"><\/span> \u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043a\u0443\u0441\u043e\u0447\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u0423\u0437\u043d\u0430\u0432 \u0441\u0432\u044f\u0437\u044c \u043c\u0435\u0436\u0434\u0443 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c\u044e \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c, \u043a\u0430\u043a \u0438\u0437\u0443\u0447\u0430\u0442\u044c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043a\u0443\u0441\u043e\u0447\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438.<\/p>\n<p> \u0412\u044b \u043c\u043e\u0436\u0435\u0442\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u043a\u0443\u0441\u043e\u0447\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u043a\u0430\u043a\u043e\u0439-\u043b\u0438\u0431\u043e \u0442\u043e\u0447\u043a\u0435, \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0432 <strong>\u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435<\/strong> \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 20px; border-radius:20px;\">\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">\u0415\u0441\u043b\u0438 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0432 \u0442\u043e\u0447\u043a\u0435 \u043d\u0435 \u0440\u0430\u0432\u043d\u044b, \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435:<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97c60c64dc01a7e0a9084313d15b0886_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x_o^-) \\neq f'(x_o^+) \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u042d\u0442\u043e \u043d\u0435 \u043f\u043e\u0434\u043b\u0435\u0436\u0438\u0442 \u0432\u044b\u0447\u0435\u0442\u0443 \u0432<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ab403a6241009b92035b251f86c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_o\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">\u0415\u0441\u043b\u0438 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0432 \u0442\u043e\u0447\u043a\u0435 \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u044e\u0442, \u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430:<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3f9477828318b9e6392465762f831642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x_o^-) = f'(x_o^+) \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u0414\u0430, \u043e\u043d\u043e \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ab403a6241009b92035b251f86c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_o\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<\/div>\n<div style=\"background-color:#FFFDE7; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> <strong>\u041f\u0440\u0438\u043c\u0435\u0447\u0430\u043d\u0438\u0435.<\/strong> \u0427\u0442\u043e\u0431\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0431\u044b\u043b\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439 \u0432 \u0442\u043e\u0447\u043a\u0435, \u043e\u043d\u0430 \u0434\u043e\u043b\u0436\u043d\u0430 \u0431\u044b\u0442\u044c \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435. \u041f\u043e\u044d\u0442\u043e\u043c\u0443, \u043f\u0440\u0435\u0436\u0434\u0435 \u0447\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435, \u043d\u0430\u043c \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0443\u0431\u0435\u0434\u0438\u0442\u044c\u0441\u044f, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435. \u0415\u0441\u043b\u0438 \u0432\u044b \u043d\u0435 \u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0438\u0437\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0432 \u0442\u043e\u0447\u043a\u0435, \u0432\u044b \u043c\u043e\u0436\u0435\u0442\u0435 \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c, \u043a\u0430\u043a \u044d\u0442\u043e \u0434\u0435\u043b\u0430\u0435\u0442\u0441\u044f, \u043f\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0441\u0441\u044b\u043b\u043a\u0435:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>\u0421\u043c.:<\/strong> <span style=\"text-decoration: underline;\"> \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u043e\u0447\u043a\u0435. <\/span><\/p>\n<\/div>\n<p> \u0422\u0435\u043f\u0435\u0440\u044c \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0435, \u043a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u043a\u0443\u0441\u043e\u0447\u043d\u043e \u0432 \u0442\u043e\u0447\u043a\u0435:<\/p>\n<ul>\n<li> \u0418\u0437\u0443\u0447\u0438\u0442\u0435 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u043a\u0443\u0441\u043e\u0447\u043d\u043e \u0432 \u0442\u043e\u0447\u043a\u0435 x=2:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a98eee72521c68fd394eb6209a7d0a59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=  \\left\\{ \\begin{array}{lcl} 3x^2-6x &amp; \\text{si} &amp;  x<2 \\\\[2ex] 6\\ln (x-1) &amp; \\text{si} &amp; x\\geq 2 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"249\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u0424\u0443\u043d\u043a\u0446\u0438\u0438 \u0434\u0432\u0443\u0445 \u0447\u0430\u0441\u0442\u0435\u0439 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b \u0432 \u0441\u0432\u043e\u0438\u0445 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u0430\u0445, \u043e\u0434\u043d\u0430\u043a\u043e \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c, \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 x=2. \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u0440\u0435\u0448\u0438\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u043e\u0447\u043a\u0435:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dad5bcb0055431aa87a67068c04d2ce2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 2^-} f(x) = \\lim\\limits_{x\\to 2^-} \\bigl(3x^2-6x\\bigr) = 3\\cdot2^2-6\\cdot2=12-12=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"449\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31e1ba2ea2c5fd9fa86e5cefed0e5535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 2^+} f(x) = \\lim\\limits_{x\\to 2^+} 6\\ln (x-1) = 6\\ln (2-1)=6 \\ln 1=6 \\cdot 0= \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"474\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> \u0411\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0434\u0430\u043b\u0438 \u043d\u0430\u043c \u0442\u043e\u0442 \u0436\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 <strong>\u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u0442\u043e\u0447\u043a\u0435 x=2.<\/strong><\/p>\n<p> \u041a\u0430\u043a \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u0443\u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u0442\u043e\u0447\u043a\u0435 x = 2, \u043c\u044b \u0438\u0437\u0443\u0447\u0438\u043c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435. \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e <strong>\u0432\u044b\u0447\u0438\u0441\u043b\u0438\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435<\/strong> \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u0432 \u043a\u0443\u0441\u043a\u0430\u0445:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3709995609d0f69f382ff651e397c00a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 6x-6 &amp; \\text{si} &amp;  x<2 \\\\[2ex] \\cfrac{6}{x-1} &amp; \\text{si} &amp; x\\geq 2 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"222\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u0422\u0435\u043f\u0435\u0440\u044c \u043c\u044b \u043e\u0446\u0435\u043d\u0438\u043c \u043a\u0430\u0436\u0434\u0443\u044e \u0431\u043e\u043a\u043e\u0432\u0443\u044e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca27189960575c1151402d040bfa76f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^-)=6\\cdot2-6=12-6 = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-058310a16d7d545ea56e99517845842b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^+)=\\cfrac{6}{2-1} = \\cfrac{6}{1} = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"181\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> \u0414\u0432\u0435 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043b\u0438 \u043d\u0430\u043c \u043e\u0434\u0438\u043d \u0438 \u0442\u043e\u0442 \u0436\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043f\u0440\u0438 x = 2, \u0430 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0440\u0430\u0432\u043d\u043e 6:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aadf046f27916c46f0a302d6e0c34113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^-) = f'(2^+) = 6 \\ \\longrightarrow \\ \\bm{f'(2) = 6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"275\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u0421 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b, \u0435\u0441\u043b\u0438 \u0431\u044b \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043b\u0438 \u043d\u0430\u043c \u0434\u0440\u0443\u0433\u043e\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442, \u044d\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u043b\u043e \u0431\u044b, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043f\u0440\u0438 x=2. \u0414\u0440\u0443\u0433\u0438\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0432 \u044d\u0442\u043e\u0442 \u043c\u043e\u043c\u0435\u043d\u0442 \u043d\u0435 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043b\u0430 \u0431\u044b.<\/p>\n<p> \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u043f\u0440\u043e\u0441\u0442\u043e \u043f\u043e\u043c\u043d\u0438\u0442\u0435, \u0447\u0442\u043e \u044d\u0442\u0430 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u0430 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u0430 \u0438 \u0434\u043b\u044f \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0430\u0431\u0441\u043e\u043b\u044e\u0442\u043d\u043e\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0430\u0431\u0441\u043e\u043b\u044e\u0442\u043d\u043e\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0442\u0430\u043a\u0436\u0435 \u043c\u043e\u0433\u0443\u0442 \u0431\u044b\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u044b \u043a\u0443\u0441\u043e\u0447\u043d\u043e. \u0417\u0434\u0435\u0441\u044c \u0432\u044b \u043c\u043e\u0436\u0435\u0442\u0435 \u0443\u0432\u0438\u0434\u0435\u0442\u044c, \u043a\u0430\u043a \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u0430\u0431\u0441\u043e\u043b\u044e\u0442\u043d\u043e\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0432 \u0444\u0440\u0430\u0433\u043c\u0435\u043d\u0442\u044b:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>\u0421\u043c.:<\/strong> <span style=\"text-decoration: underline;\"> \u043a\u0430\u043a \u043a\u0443\u0441\u043e\u0447\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u0441 \u0430\u0431\u0441\u043e\u043b\u044e\u0442\u043d\u044b\u043c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435\u043c <\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivabilidad-de-una-funcion\"><\/span> \u0420\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u043d\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> \u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 1<\/h3>\n<p> \u0418\u0437\u0443\u0447\u0438\u0442\u0435 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u043a\u0443\u0441\u043e\u0447\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3656065bb8de98bd07da153f26fd326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^3-4x^2 + 5 &amp; \\text{si} &amp;  x<1 \\\\[2ex] -x^2+3x &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"265\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u0424\u0443\u043d\u043a\u0446\u0438\u0438 \u0434\u0432\u0443\u0445 \u0447\u0430\u0441\u0442\u0435\u0439 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b, \u043d\u043e \u043c\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c, \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 x=1. \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u0440\u0435\u0448\u0438\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u043e\u0447\u043a\u0435: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf8ad0b4baa312a5ae1bb073e5c8ff8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^-} f(x) = \\lim\\limits_{x\\to 1^-} \\bigl(x^3-4x^2 + 5\\bigr)=1^3-4\\cdot 1^2 + 5=2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"413\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f25bd83439fbb5cbd148b8be88f8770b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^+} f(x) = \\lim\\limits_{x\\to 1^+} \\bigl( -x^2+3x \\bigr)=-1^2+3\\cdot 1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"364\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0414\u0432\u0430 \u0431\u043e\u043a\u043e\u0432\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0434\u0430\u044e\u0442 \u043e\u0434\u0438\u043d \u0438 \u0442\u043e\u0442 \u0436\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u043f\u0440\u0438 x=1.<\/p>\n<p class=\"has-text-align-left\"> \u041a\u0430\u043a \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u0443\u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435, \u043c\u044b \u0443\u0437\u043d\u0430\u0435\u043c, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u043e\u043d\u0430 \u0432 \u0442\u043e\u0439 \u0436\u0435 \u0442\u043e\u0447\u043a\u0435. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0435\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42451fa799527167fe9a2e2259248870_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 3x^2-8x  &amp; \\text{si} &amp;  x<1 \\\\[2ex] -2x+3 &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"240\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0418 \u043c\u044b \u043e\u0446\u0435\u043d\u0438\u0432\u0430\u0435\u043c \u0434\u0432\u0435 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043f\u0440\u0438 x=1; <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9afcac5ff2f762d2b471725d4e755fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-)=3\\cdot1^2-8\\cdot 1=3-8=-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"272\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dd2d9e21e09ecc434cafbf5fb0b5afaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^+)=-2\\cdot 1+3=-2+3 =1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0411\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043d\u0435 \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u044e\u0442 \u0432 \u0442\u043e\u0447\u043a\u0435 x=1, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73b9cb3dada6f8aea03ffc1342d0f22f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-) \\neq f'(1^+) \\ \\longrightarrow \\ \\cancel{\\exists} \\ f'(1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 2<\/h3>\n<p> \u041f\u0440\u043e\u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0443\u0439\u0442\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u0432 \u0440\u0430\u0437\u0434\u0435\u043b\u0430\u0445: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d118e3904c810abd15e427e9c7d0504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} \\sqrt{4x} &amp; \\text{si} &amp;  x\\leq 1 \\\\[2ex] 2+\\ln x &amp; \\text{si} &amp; x> 1 \\end{array} \\right.&#187; title=&#187;Rendered by QuickLaTeX.com&#187; height=&#187;65&#8243; width=&#187;226&#8243; style=&#187;vertical-align: 0px;&#187;><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u0424\u0443\u043d\u043a\u0446\u0438\u0438 \u0434\u0432\u0443\u0445 \u0443\u0447\u0430\u0441\u0442\u043a\u043e\u0432 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b \u043d\u0430 \u0441\u0432\u043e\u0438\u0445 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u0430\u0445, \u043d\u043e \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0442\u0430\u043a\u0436\u0435 \u0437\u043d\u0430\u0442\u044c, \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f x=1. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043c\u044b \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0435\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c4fb7ef0ee9b3feeb5e15654528fc71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^-} f(x) = \\lim\\limits_{x\\to 1^-} \\sqrt{4x} = \\sqrt{4\\cdot 1} = \\sqrt{4}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"323\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e509a70bd0e356f44ccac7ba6f075f8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^+} f(x) = \\lim\\limits_{x\\to 1^+} \\bigl( 2+\\ln x \\bigr) = 2 + \\ln (1) = 2+0 =2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"399\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0414\u0432\u0430 \u0431\u043e\u043a\u043e\u0432\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0434\u0430\u044e\u0442 \u043e\u0434\u0438\u043d \u0438 \u0442\u043e\u0442 \u0436\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u043f\u0440\u0438 x=1.<\/p>\n<p class=\"has-text-align-left\"> \u0410 \u0442\u0435\u043f\u0435\u0440\u044c \u0443\u0437\u043d\u0430\u0435\u043c, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435, \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0432 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8261f3d268b47d9171710997c8cc70bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} \\cfrac{4}{2\\sqrt{4x}}  &amp; \\text{si} &amp;  x<1 \\\\[4ex] \\cfrac{1}{x} &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"217\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u041c\u044b \u043e\u0446\u0435\u043d\u0438\u0432\u0430\u0435\u043c \u0434\u0432\u0435 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043f\u0440\u0438 x=1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de4ee71a175b34be07061d47470afe0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-)=\\cfrac{4}{2\\sqrt{4\\cdot1}}=\\cfrac{4}{2\\sqrt{4}}=\\cfrac{4}{2\\cdot 2}=\\cfrac{4}{4}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"309\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-594a8d05da432ab8962ff799d62d25a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^+)=\\cfrac{1}{1}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"115\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0411\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0440\u0430\u0432\u043d\u044b, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043f\u0440\u0438 x = 1, \u0430 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u0440\u0430\u0432\u043d\u043e 1. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a58ba22b03193839f5070da4e2c1faf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-) = f'(1^+) = 1 \\ \\longrightarrow \\ \\bm{f'(1) = 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"274\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 3<\/h3>\n<p> \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u0435, \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043b\u0438 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u043a\u0443\u0441\u043e\u0447\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439 \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439 \u0432\u043e \u0432\u0441\u0435\u0439 \u0441\u0432\u043e\u0435\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f:<\/p>\n<pre class=\"ql-errors\"> *** QuickLaTeX cannot compile formula:\n\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^2+2x+1 &amp; \\text{si} &amp; x\\leq -1 \\\\[2ex] 2x+2 &amp; \\text{ si} &amp; -1&lt;div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria- expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\"&gt;&lt;div class=\"otfm-sp__title\"&gt; &lt;strong&gt;View solution&lt;\/strong&gt;&lt;\/div&gt;&lt; \/div&gt; The functions of all three parts are continuous, but we still need to check if the function is continuous at critical points. We therefore first check the continuity of the function at the point x=-1 by solving the lateral limits at this point:\n\n*** Error message:\nMissing $ inserted.\nleading text: \\displaystyle\nMissing { inserted.\nleading text: ...=\"wp-block-otfm-box-spoiler-start otfm-sp__\nMissing { inserted.\nleading text: ...ox-spoiler-start otfm-sp__wrapper otfm-sp__\nMissing { inserted.\nleading text: ...m-sp__wrapper otfm-sp__box js-otfm-sp-box__\nMissing { inserted.\nleading text: ...fm-sp__box js-otfm-sp-box__closed otfm-sp__\nYou can't use `macro parameter character #' in math mode.\nleading text: ...=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#\nMissing { inserted.\nleading text: ...e=\"text-align:center\"&gt;&lt;div class=\"otfm-sp__\nPlease use \\mathaccent for accents in math mode.\nleading text: ...g&gt;&lt;\/div&gt;&lt;\/div&gt; The functions of the three parts\nPlease use \\mathaccent for accents in math mode.\nleading text: ...are continuous, but we still need to see\n\n<\/pre>\n<p> \\lim\\limits_{x\\to -1^-} f(x) = \\lim\\limits_{x\\to -1^-} \\bigl(x^2+2x+1\\bigr) = (-1)^ 2+2(-1)+1 =0 \\lim\\limits_{x\\to -1^+} f(x) = \\lim\\limits_{x\\to -1^+} \\bigl(2x+2\\bigr ) = 2(-1)+2=0<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fadd0ce26a497a6a6c73bfaa7ed28f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" Les deux limites lat\u00e9rales au point x=-1 donnent le m\u00eame r\u00e9sultat, donc la fonction est continue en x=-1. Nous allons maintenant v\u00e9rifier si la fonction est continue ou non au point x=2 : \" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"765\" style=\"vertical-align: 0px;\"><\/p>\n<p> \\lim\\limits_{x\\to 2^-} f(x) = \\lim\\limits_{x\\to 2^-} \\bigl(2x+2\\bigr) = 2\\cdot 2+2=4+2= 6 \\lim\\limits_{x\\to 2^+} f(x) = \\lim\\limits_{x\\to 2^+} \\bigl( -x^2+8x\\bigr) = -2^2+8\\ CDOT 2 = -4+16=12<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c93f9d5e367031de798abaf833523710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" En revanche, les limites lat\u00e9rales au point x=2 ne donnent pas le m\u00eame r\u00e9sultat, donc la fonction n'est pas continue en x=2. De plus, comme il n'est pas continu \u00e0 ce stade, il ne sera pas non plus d\u00e9rivable \u00e0 x=2. Une fois que l'on a \u00e9tudi\u00e9 la continuit\u00e9 de la fonction, on passe \u00e0 la diff\u00e9rentiabilit\u00e9. On calcule donc les d\u00e9riv\u00e9es lat\u00e9rales :\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"908\" style=\"vertical-align: -4px;\"><\/p>\n<p> \\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 2x+2 &amp; \\text{si} &amp; x\\leq -1 \\\\[2ex] 2 &amp; \\text{si} &amp; -1<\/p>\n<p class=\"has-text-align-left\"> \u041c\u044b \u0443\u0436\u0435 \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043f\u0440\u0438 x=2, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043d\u0430\u043c \u043f\u0440\u043e\u0441\u0442\u043e \u043d\u0443\u0436\u043d\u043e \u0438\u0437\u0443\u0447\u0438\u0442\u044c, \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043f\u0440\u0438 x=-1. \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u043e\u0446\u0435\u043d\u0438\u043c \u0434\u0432\u0435 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0432 \u0442\u043e\u0447\u043a\u0435: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cafb8bbe438865e050973664b6915fa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^-)=2(-1)+2 = -2+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"272\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5781d213b36777be5b2611a84ca95e96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^+)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0411\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u043d\u0435 \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u044e\u0442 \u0432 \u0442\u043e\u0447\u043a\u0435 x=-1, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u043d\u0435 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3674d70c292d967d7074a0b4bee230e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^-) \\neq f'(-1^+) \\ \\longrightarrow \\ \\cancel{\\exists} \\ f'(-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"267\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0435 4<\/h3>\n<p> \u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u043e\u0432 a \u0438 b \u0442\u0430\u043a, \u0447\u0442\u043e\u0431\u044b \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u043a\u0443\u0441\u043e\u0447\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0431\u044b\u043b\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439 \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439 \u0432\u043e \u0432\u0441\u0435\u0439 \u0441\u0432\u043e\u0435\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce34d5d8a949fb3a0b904e9bf7d32f5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} 2e^{x-3} + a &amp; \\text{si} &amp;  x< 3 \\\\[2ex](x-b)^2 &amp; \\text{si} &amp; x\\geq 3 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"243\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0435\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u041a\u0430\u043a\u0438\u043c\u0438 \u0431\u044b \u043d\u0438 \u0431\u044b\u043b\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u043d\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b\u0445, \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432\u043e \u0432\u0441\u0435\u0445 \u0442\u043e\u0447\u043a\u0430\u0445, \u043a\u0440\u043e\u043c\u0435 \u0442\u043e\u0447\u043a\u0438 x=3, \u0433\u0434\u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c \u0435\u0435 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c.<\/p>\n<p class=\"has-text-align-left\"> \u0427\u0442\u043e\u0431\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0431\u044b\u043b\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439 \u0432 \u0442\u043e\u0447\u043a\u0435, \u0434\u0432\u0430 \u0431\u043e\u043a\u043e\u0432\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0432 \u044d\u0442\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 \u0434\u043e\u043b\u0436\u043d\u044b \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0442\u044c. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043e\u0446\u0435\u043d\u0438\u0432\u0430\u0435\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21920afd0fe6c6a35983a39034b3f9f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 3^-} f(x) = \\lim\\limits_{x\\to 3^-} \\bigl(2e^{x-3}+a\\bigr) = 2e^{3-3}+a = 2 \\cdot e^0+a =2\\cdot 1 +a = 2+a\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"566\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f527d4602a1bdc26d763df1064b956d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 3^+} f(x) = \\lim\\limits_{x\\to 3^+} (x-b)^2 = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"282\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0434\u0432\u0430 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u0438\u0437 \u0431\u043e\u043a\u043e\u0432\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432, \u0434\u043e\u043b\u0436\u043d\u044b \u0431\u044b\u0442\u044c \u0440\u0430\u0432\u043d\u044b, \u0447\u0442\u043e\u0431\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0431\u044b\u043b\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0439:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3bec8303f39289b7f2cd7e6e439703c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0422\u0435\u043f\u0435\u0440\u044c \u043f\u0440\u043e\u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0443\u0435\u043c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0432 \u0442\u043e\u0447\u043a\u0435 x=3. \u041d\u0430\u0445\u043e\u0434\u0438\u043c \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d542fc9488644f0c144059ae1403d961_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 2e^{x-3}  &amp; \\text{si} &amp;  x< 3 \\\\[2ex]2(x-b) &amp; \\text{si} &amp; x\\geq 3 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"236\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0418 \u043c\u044b \u043e\u0446\u0435\u043d\u0438\u0432\u0430\u0435\u043c \u0434\u0432\u0435 \u0431\u043e\u043a\u043e\u0432\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0432 \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0435: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0fb3afef2352f0f5f96302b987a5de9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3^-)= 2e^{3-3} =  2e^0 = 2\\cdot 1 = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"250\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-360474b477f347569ad1e3b64b63cc79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3^+)=2(3-b) = 6 - 2b\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0447\u0442\u043e\u0431\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0431\u044b\u043b\u0430 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0439 \u043f\u0440\u0438 x=3, \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u043e\u0442 \u0431\u043e\u043a\u043e\u0432\u044b\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445, \u0434\u043e\u043b\u0436\u043d\u044b \u0431\u044b\u0442\u044c \u0440\u0430\u0432\u043d\u044b:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a98a45230845ad86846cd7db486af0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=6-2b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u0420\u0435\u0448\u0438\u0432 \u044d\u0442\u043e \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043d\u0430\u0439\u0442\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 b: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f53d30cae8270029d25ea28322b6986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2b=6-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c263311ac7433ce2b417bb7ad0ef449b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2b=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7dd2f0fd5e5207a815bf5789ada67541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=\\cfrac{4}{2} =\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> \u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u043a\u0430\u043a \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u0443\u0437\u043d\u0430\u0435\u043c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 b, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 a, \u0440\u0435\u0448\u0438\u0432 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0440\u0430\u043d\u0435\u0435 \u0432 \u0431\u043e\u043a\u043e\u0432\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u0430\u0445: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3bec8303f39289b7f2cd7e6e439703c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6468721c373f078ad3e97a290c2d86f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-2)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1cc01602cd205cae7b9c9b8ba391760_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a =1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"72\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce22cc1bba3ee9612a7f8cb2624d2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a =1-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"72\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be5f9e4b074e8b3ba5d0e96f8ae4e2cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a =-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u0412 \u044d\u0442\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u0432\u044b \u0443\u0437\u043d\u0430\u0435\u0442\u0435, \u043a\u0430\u043a \u0438\u0437\u0443\u0447\u0430\u0442\u044c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0442\u043e \u0435\u0441\u0442\u044c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0438\u043b\u0438 \u043d\u0435\u0442. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c \u0441\u0432\u044f\u0437\u044c \u043c\u0435\u0436\u0434\u0443 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c\u044e \u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0418, \u043d\u0430\u043a\u043e\u043d\u0435\u0446, \u0438\u0437\u0443\u0447\u0438\u043c \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u043a\u0443\u0441\u043e\u0447\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u041d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u044c \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u043e\u0447\u043a\u0435 \u0441\u0432\u044f\u0437\u0430\u043d\u044b \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c: \u0415\u0441\u043b\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u0442\u043e\u0447\u043a\u0435, \u0442\u043e \u043e\u043d\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0432 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/ru\/%d0%b4%d0%b8%d1%84%d1%84%d0%b5%d1%80%d0%b5%d0%bd%d1%86%d0%b8%d1%80%d1%83%d0%b5%d0%bc%d0%be%d1%81%d1%82%d1%8c-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8\/\"> <span class=\"screen-reader-text\">\u0414\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/span> \u0427\u0438\u0442\u0430\u0442\u044c \u0434\u0430\u043b\u0435\u0435 &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[6],"tags":[],"class_list":["post-41","post","type-post","status-publish","format-standard","hentry","category-6"],"yoast_head":"<!-- This site is optimized 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