{"id":1009,"date":"2023-08-06T16:12:53","date_gmt":"2023-08-06T13:12:53","guid":{"rendered":"https:\/\/bagrut-ru.com\/?p=1009"},"modified":"2025-09-04T17:43:41","modified_gmt":"2025-09-04T14:43:41","slug":"%d0%b1%d0%b0%d0%b3%d1%80%d1%83%d1%82-5-%d0%bb%d0%b5%d1%82%d0%be-2023-%d0%b2%d0%b0%d1%80%d0%b8%d0%b0%d0%bd%d1%82-35582-%d0%b7%d0%b0%d0%b4%d0%b0%d0%bd%d0%b8%d0%b5-5","status":"publish","type":"post","link":"https:\/\/bagrut-ru.com\/?p=1009","title":{"rendered":"\u0411\u0410\u0413\u0420\u0423\u0422 (5). \u041b\u0415\u0422\u041e 2023, \u0412\u0410\u0420\u0418\u0410\u041d\u0422 35582, \u0417\u0410\u0414\u0410\u041d\u0418\u0415 5"},"content":{"rendered":"\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"501\" data-id=\"1010\" src=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/90-1024x501.png\" alt=\"\" class=\"wp-image-1010\" srcset=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/90-1024x501.png 1024w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/90-300x147.png 300w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/90-768x375.png 768w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/90.png 1264w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/figure>\n\n\n\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-2 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"555\" data-id=\"1011\" src=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/91-1024x555.png\" alt=\"\" class=\"wp-image-1011\" srcset=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/91-1024x555.png 1024w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/91-300x163.png 300w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/91-768x416.png 768w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/91.png 1384w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/meyda.education.gov.il\/sheeloney_bagrut\/2023\/6\/RUS\/35582.pdf\" data-type=\"link\" data-id=\"https:\/\/meyda.education.gov.il\/sheeloney_bagrut\/2023\/6\/RUS\/35582.pdf\">\u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a<\/a><\/p>\n\n\n<div class=\"spoiler-wrap\">\n\t\t\t\t<div class=\"spoiler-head folded\">\u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/div>\n\t\t\t\t<div class=\"spoiler-body\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">(\u05d0)<\/mark><\/strong> \u0414\u043b\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(f(x)=\\ln(x)+\\frac{1}{x}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>(1)<\/strong><\/mark> \u041d\u0430\u0439\u0434\u0435\u043c \u043e\u0431\u0440\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(\\ln x: \\ x&gt;0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(\\frac{1}{x}: \\ x\\ne0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(f(x)=\\ln(x)+\\frac{1}{x}: x&gt;0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>\u041e\u0442\u0432\u0435\u0442:<\/em> \\(x&gt;0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(2)<\/mark> <\/strong>\u041d\u0430\u0439\u0434\u0435\u043c \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0442\u043e\u0447\u043a\u0438 \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u0430.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041d\u0430\u0439\u0434\u0435\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(f(x)=\\ln(x)+\\frac{1}{x}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f'(x)=(\\ln(x)+\\frac{1}{x})&#8217;=\\frac{1}{x}-\\frac{1}{x^2}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u0440\u0438\u0440\u0430\u0432\u043d\u044f\u0435\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a \u043d\u0443\u043b\u044e:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\frac{1}{x}-\\frac{1}{x^2}=0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\frac{x-1}{x^2}=0\\) <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0414\u0440\u043e\u0431\u044c \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e, \u043a\u043e\u0433\u0434\u0430 \u0447\u0438\u0441\u043b\u0438\u0442\u0435\u043b\u044c \u0440\u0430\u0432\u0435\u043d \u043d\u0443\u043b\u044e, \u0430 \u0437\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u044c \u043d\u0435 \u0440\u0430\u0432\u0435\u043d. \\(x-1=0, \\ x=1\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0415\u0441\u043b\u0438 \\(x&gt;1,\\ f'(x)=\\frac{x-1}{x^2}&gt;0\\), \u0435\u0441\u043b\u0438 \\(x&lt;1,\\ f'(x)=\\frac{x-1}{x^2}&lt;0\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0432 \u0442\u043e\u0447\u043a\u0435 \\( x=1\\) \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043c\u0435\u043d\u044f\u0435\u0442 \u0437\u043d\u0430\u043a \u0441 &#171;-&#187; \u043d\u0430 &#171;+&#187;. \u041e\u0442\u0441\u044e\u0434\u0430 \u0442\u043e\u0447\u043a\u0430 \\( x=1\\) &#8212; \u0442\u043e\u0447\u043a\u0430 \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u0430. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f(1)=\\ln(1)+\\frac{1}{1}=1\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>\u041e\u0442\u0432\u0435\u0442: <\/em>\\(min(1,1)\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(3)<\/mark><\/strong> \u0412 \u043f. (2) \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438, \u0447\u0442\u043e \\(f'(x)&lt;0 \\), \u0435\u0441\u043b\u0438  \\(x&lt;1\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043d\u0430 \u044d\u0442\u043e\u043c \u043f\u0440\u043e\u043c\u0435\u0436\u0443\u0442\u043a\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \\(f(x)=\\ln(x)+\\frac{1}{x}\\) \u0443\u0431\u044b\u0432\u0430\u0435\u0442. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f'(x)&gt;0 \\), \u0435\u0441\u043b\u0438  \\(x&gt;1\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043d\u0430 \u044d\u0442\u043e\u043c \u043f\u0440\u043e\u043c\u0435\u0436\u0443\u0442\u043a\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \\(f(x)=\\ln(x)+\\frac{1}{x}\\) \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u0435\u0442. <\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"wp-block-paragraph\">\u041f\u0440\u0438 \\(x\\rightarrow 0\\ \\)  \\(ln x\\rightarrow \\infty\\) \u0438 \\(\\frac{1}{x}\\rightarrow \\infty\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043f\u0440\u0438 \\(x\\rightarrow 0\\ \\)  \\(f(x)\\rightarrow 0\\) \u0438 \u043e\u0442\u0441\u044e\u0434\u0430 \\(x=0\\) &#8212; \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u044c\u043d\u0430\u044f \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0430. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0423\u0447\u0438\u0442\u044b\u0432\u0430\u044f \u043f. (1) \u0438 (2) \u0438\u0437\u043e\u0431\u0440\u0430\u0437\u0438\u043c \u0441\u0445\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u043e \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438. <\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-3 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"663\" data-id=\"1015\" src=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/92-1024x663.png\" alt=\"\" class=\"wp-image-1015\" srcset=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/92-1024x663.png 1024w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/92-300x194.png 300w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/92-768x497.png 768w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/92.png 1124w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">(\u05d1)<\/mark><\/strong> \u0414\u043b\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(g(x)=(x+1)(1-\\ln x)\\), \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u043f\u0440\u0438 \\(x&gt;0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(1)<\/mark><\/strong> \u041d\u0430\u0439\u0434\u0435\u043c \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0442\u043e\u0447\u043a\u0438 \u043f\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(g(x)\\) \u0441 \u043e\u0441\u044c\u044e \\(x\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430 \\(y\\) \u0442\u043e\u0447\u0435\u043a \u043f\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0441 \u043e\u0441\u044c\u044e \\(x\\) \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e. \u0427\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \\(x\\) \u0442\u043e\u0447\u0435\u043a \u043f\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043d\u0438\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(g(x)\\) \u0441 \u043e\u0441\u044c\u044e \\(x\\), \u0440\u0435\u0448\u0438\u043c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\((x+1)(1-\\ln x)=0\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439 \u0440\u0430\u0432\u043d\u043e \u043d\u0443\u043b\u044e, \u0435\u0441\u043b\u0438 \u0445\u043e\u0442\u044f \u0431\u044b \u043e\u0434\u0438\u043d \u0438\u0437 \u043d\u0438\u0445 \u0440\u0430\u0432\u0435\u043d \u043d\u0443\u043b\u044e, \u0430 \u0432\u0442\u043e\u0440\u043e\u0439 \u043f\u0440\u0438 \u044d\u0442\u043e\u043c \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(x+1=0, \\ x=-1\\) \u041d\u043e \\(\\ln x\\) \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d \u043f\u0440\u0438 \\(x=-1\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \\(x=-1\\) &#8212; \u043f\u043e\u0441\u0442\u043e\u0440\u043e\u043d\u043d\u0438\u0439 \u043a\u043e\u0440\u0435\u043d\u044c.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(1-\\ln x=0,\\ \\ln x=1,\\ x=e\\) <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041e\u0442\u0432\u0435\u0442: (\\(e,0)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(2)<\/mark><\/strong> \u041d\u0430\u0439\u0434\u0435\u043c \u043f\u0440\u043e\u043c\u0435\u0436\u0443\u0442\u043a\u0438 \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u043d\u0438\u044f \u0438 \u0443\u0431\u044b\u0432\u0430\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(g(x)=(x+1)(1-\\ln x)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041d\u0430\u0439\u0434\u0435\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u043f\u0440\u043e\u0438\u0432\u0435\u0434\u0435\u043d\u0438\u044f:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">\\((UV)&#8217;=U&#8217;V+V&#8217;U\\)<\/mark><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0417\u0434\u0435\u0441\u044c \\(U=x+1,\\ U&#8217;=1,\\ V=1-\\ln x,\\ V&#8217;=-\\frac{1}{x}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0422\u043e\u0433\u0434\u0430 \\(g'(x)=1\\cdot (1-\\ln x)+(-\\frac{1}{x})(x+1)=1-\\ln x-1-\\frac{1}{x}=-\\ln x-\\frac{1}{x}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u0440\u0438\u0440\u0430\u0432\u043d\u044f\u0435\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a \u043d\u0443\u043b\u044e. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(  g'(x)=-\\ln x-\\frac{1}{x}=0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \\(g'(x)=-f(x)\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(f(x)&gt;0\\) \u043f\u0440\u0438 \u043b\u044e\u0431\u043e\u043c \\(x&gt;0\\) (\u0441\u043c \u043f. <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">(\u05d0)<\/mark><\/strong>), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \\(g'(x)&lt;0\\) \u043f\u0440\u0438 \u043b\u044e\u0431\u043e\u043c \\(x&gt;0\\). \u041e\u0442\u0441\u044e\u0434\u0430 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \\(g(x)\\) \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0442\u043e\u0447\u0435\u043a \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u0430 \u0438 \u0443\u0431\u044b\u0432\u0430\u0435\u0442 \u043d\u0430 \u0432\u0441\u0435\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041e\u0442\u0432\u0435\u0442: \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0443\u0431\u044b\u0432\u0430\u0435\u0442 \u043f\u0440\u0438 \\(x&gt;0\\). <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(3)<\/mark><\/strong> \u041d\u0430\u0439\u0434\u0435\u043c \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u0432\u043e\u0433\u043d\u0443\u0442\u043e\u0441\u0442\u0438 \u0432\u0432\u0435\u0440\u0445 \\((\\cup)\\) \u0438 \u0432\u043e\u0433\u043d\u0443\u0442\u043e\u0441\u0442\u0438 \u0432\u043d\u0438\u0437 \\( (\\cap)\\) \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(g(x)=(x+1)(1-\\ln x)\\). \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u043d\u0430\u0439\u0434\u0435\u043c \u043f\u0440\u043e\u043c\u0435\u0436\u0443\u0442\u043a\u0438 \u0437\u043d\u0430\u043a\u043e\u043f\u043e\u0441\u0442\u043e\u044f\u043d\u0441\u0442\u0432\u0430 \u0432\u0442\u043e\u0440\u043e\u0439 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439 \u044d\u0442\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(  g'(x)=-\\ln x-\\frac{1}{x}\\) (\u0438\u0437 <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(2)<\/mark><\/strong> )<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(  (g'(x))&#8217;=(-\\ln x-\\frac{1}{x})&#8217;=-\\frac{1}{x}+\\frac{1}{x^2}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u0440\u0438\u0440\u0430\u0432\u043d\u044f\u0435\u043c \u0432\u0442\u043e\u0440\u0443\u044e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0443\u044e \u043a \u043d\u0443\u043b\u044e. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(-\\frac{1}{x}+\\frac{1}{x^2}=0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\frac{1-x}{x^2}=0\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0412\u0442\u043e\u0440\u0430\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e \u043f\u0440\u0438 \\(x=1\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u0440\u0438  \\(x&gt;1\\) \\(\\frac{1-x}{x^2}&lt;0\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e \u0432\u0442\u043e\u0440\u0430\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430 \u0438 \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432\u043e\u0433\u043d\u0443\u0442 \u0432\u043d\u0438\u0437 \\( (\\cap)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u0440\u0438  \\(x&lt;1\\) \\(\\frac{1-x}{x^2}&gt;0\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e \u0432\u0442\u043e\u0440\u0430\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430 \u0438 \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432\u043e\u0433\u043d\u0443\u0442 \u0432\u0432\u0435\u0440\u0445 \\((\\cup)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">  \\(x=1\\) &#8212; \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430 \\(x\\) \u0442\u043e\u0447\u043a\u0438 \u043f\u0435\u0440\u0435\u0433\u0438\u0431\u0430.  \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430 \\( y\\) \u0442\u043e\u0447\u043a\u0438 \u043f\u0435\u0440\u0435\u0433\u0438\u0431\u0430: \\(g(1)=(1+1)(1-\\ln 1)=2\\)<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"wp-block-paragraph\">\u0424\u0443\u043d\u043a\u0446\u0438\u044f \\(g(x)=(x+1)(1-\\ln x)\\) \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u043f\u0440\u0438 \\(x&gt;0\\). \u0415\u0441\u043b\u0438 \\(x\\rightarrow 0\\),  \u0442\u043e \\(x+1 \\rightarrow 1\\), \\(-\\ln x\\rightarrow \\infty\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \\(g(x)\\rightarrow\\infty\\). \u0422\u043e \u0435\u0441\u0442\u044c  \\(x=0\\) &#8212; \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u044c\u043d\u0430\u044f \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0430. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0421 \u0443\u0447\u0435\u0442\u043e\u043c \u043f (1), (2), (3) \u043c\u043e\u0436\u0435\u043c \u043d\u0430\u0447\u0435\u0440\u0442\u0438\u0442\u044c \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \\(g(x)=(x+1)(1-\\ln x)\\): <\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-4 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"717\" data-id=\"1034\" src=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/93-1024x717.png\" alt=\"\" class=\"wp-image-1034\" srcset=\"https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/93-1024x717.png 1024w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/93-300x210.png 300w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/93-768x538.png 768w, https:\/\/bagrut-ru.com\/wp-content\/uploads\/2023\/08\/93.png 1256w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">(\u05d2)<\/mark><\/strong> \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044e \\(h(x)=\\frac{1}{x}\\cdot g'(x)\\), \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u0443\u044e \u043f\u0440\u0438 \\(x&gt;0\\). <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> \u0412 \u043f. <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">(\u05d1)<\/mark><\/strong> <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">(2)<\/mark><\/strong> \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438, \u0447\u0442\u043e \\(  g'(x)=-\\ln x-\\frac{1}{x}\\), \u0442\u043e\u0433\u0434\u0430 \\(h(x)=\\frac{1}{x}\\cdot (-\\ln x-\\frac{1}{x})\\), <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0412\u044b\u0447\u0438\u0441\u043b\u0438\u043c \u043f\u043b\u043e\u0449\u0430\u0434\u044c \u0444\u0438\u0433\u0443\u0440\u044b, \u0437\u0430\u043a\u043b\u044e\u0447\u0435\u043d\u043d\u043e\u0439 \u043c\u0435\u0436\u0434\u0443 \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u0438  \\(h(x)\\), \u043e\u0441\u044c\u044e \\(x\\) \u0438 \u043f\u0440\u044f\u043c\u044b\u043c\u0438 \\(x=1, \\ x=e\\). <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041f\u043b\u043e\u0449\u0430\u0434\u044c \u043d\u0430\u0445\u043e\u0434\u0438\u043c \u043f\u043e \u0444\u043e\u0440\u043c\u0443\u043b\u0435:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> \\(S=|\\int_1^e (\\frac{1}{x}\\cdot (-\\ln x-\\frac{1}{x}))dx| =\\int_1^e (\\frac{1}{x}\\cdot (\\ln x+\\frac{1}{x}))dx=\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(=\\int_1^e (\\frac{1}{x}\\cdot \\ln x )dx+\\int_1^e (\\frac{1}{x}\\cdot \\frac{1}{x})dx\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u041d\u0430\u0439\u0434\u0435\u043c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0430 \u043f\u043e \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\int_1^e (\\frac{1}{x}\\cdot \\ln x) dx\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0412\u0432\u0435\u0434\u0435\u043c \u0437\u0430\u043c\u0435\u043d\u0443: \\(u=\\ln x,\\ du=(\\ln x)&#8217;dx= \\frac{1}{x}dx, \\ \\ln 1=0, \\ln e=1\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0422\u043e\u0433\u0434\u0430 \\(\\int_1^e (\\frac{1}{x}\\cdot \\ln x )dx= \\int_0^1udu=\\frac{u^2}{2}|_0^1=\\frac{1}{2}\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\int_1^e (\\frac{1}{x}\\cdot \\frac{1}{x}dx)=\\int_1^e\\frac{1}{x^2}dx=-\\frac{1}{x}|_1^e=-\\frac{1}{e}+1\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0418\u0442\u0430\u043a: \\(S=\\frac{1}{2}-\\frac{1}{e}+1=\\frac{3}{2}-\\frac{1}{e}\\approx 1.13\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>\u041e\u0442\u0432\u0435\u0442:<\/em> \\(\\frac{3}{2}-\\frac{1}{e}\\approx 1.13\\)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/div>\n\t\t\t<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,9],"tags":[],"class_list":["post-1009","post","type-post","status-publish","format-standard","hentry","category-8","category-9"],"_links":{"self":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts\/1009","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1009"}],"version-history":[{"count":35,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts\/1009\/revisions"}],"predecessor-version":[{"id":1532,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts\/1009\/revisions\/1532"}],"wp:attachment":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}