{"id":4370,"date":"2025-10-20T19:55:10","date_gmt":"2025-10-20T11:55:10","guid":{"rendered":"http:\/\/47.95.222.19\/?p=4370"},"modified":"2025-11-15T23:05:10","modified_gmt":"2025-11-15T15:05:10","slug":"%e5%87%bd%e6%95%b0%e5%af%bc%e6%95%b03%e8%ae%b2%e8%a7%a3","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=4370","title":{"rendered":"\u51fd\u6570\u5bfc\u65703\u8bb2\u89e3"},"content":{"rendered":"\n<div class=\"wp-block-stackable-subtitle stk-block-subtitle stk-block stk-8afe3d6\" data-block-id=\"8afe3d6\"><style>.stk-8afe3d6 .stk-block-subtitle__text{font-size:var(--stk--preset--font-size--medium, 20px) !important;color:#0693e3 !important;}@media screen and (max-width:999px){.stk-8afe3d6 .stk-block-subtitle__text{font-size:var(--stk--preset--font-size--medium, 20px) !important;}}<\/style><p class=\"stk-block-subtitle__text stk-subtitle has-text-color\">\u672c\u9898\u75312023\u7ea725\u73ed\u6210\u5141\u822a\u8bb2\u89e3\uff01<\/p><\/div>\n\n\n\n<figure class=\"wp-block-video\"><video height=\"544\" style=\"aspect-ratio: 960 \/ 544;\" width=\"960\" controls poster=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/10\/\u51fd\u6570\u5bfc\u65703\u5c01\u9762.png\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/11\/\u51fd\u6570\u5bfc\u65703.1.mp4\"><\/video><\/figure>\n\n\n\n<p>[ratemypost]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-vivid-cyan-blue-color has-alpha-channel-opacity has-vivid-cyan-blue-background-color has-background is-style-wide\"\/>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-8a052dbd5caa870510f1e594e9e095cf\">\u9898\u76ee\u5c55\u793a\u2014\u2014\u5148\u63a2\u7a76\u3001\u518d\u542c\u8bb2<\/h5>\n\n\n\n<h5>\u9898\u76ee3<\/h5>\n<p>\u5df2\u77e5\u51fd\u6570\\( f(x)=x(1-\\ln x) \\)\u7684\u5355\u8c03\u6027\uff1b<\/p>\n<p>(1) \u8ba8\u8bba\\( f(x) \\)\u7684\u5355\u8c03\u6027\uff1b<\/p>\n<p>(2) \u8bbe\\( a, b \\)\u4e3a\u4e24\u4e2a\u4e0d\u76f8\u7b49\u7684\u6b63\u6570\uff0c\u4e14\\( b\\ln a &#8211; a\\ln b = a &#8211; b \\)\uff0c\u8bc1\u660e\uff1a\\( 2 <\\displaystyle \\frac{1}{a} + \\frac{1}{b} < e \\)\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-text-color has-background has-link-color has-medium-font-size wp-elements-9d0540021c94136dca84d09cf7cfb04a\" style=\"color:#1e4b69;background:linear-gradient(0deg,rgb(230,242,255) 0%,rgb(230,242,255) 16%,rgb(92,175,230) 100%)\">\u89e3\u9898\u63a2\u7a76\u2014\u2014\u70b9\u51fb\u76f4\u8fbe<\/h4>\n\n\n\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left;\n    background-color: #e6f2ff; \/* \u6de1\u84dd\u8272\u80cc\u666f *\/\n    border: none;\n    padding: 8px 12px;\n    border-radius: 4px;\n    transition: all 0.3s ease; \/* \u5e73\u6ed1\u8fc7\u6e21\u52a8\u753b *\/\n    font-size: 16px;\n  }\n  \n  .video-jump-btn:hover {\n    background-color: #cce0ff; \/* \u7565\u6df1\u7684\u84dd\u8272\u80cc\u666f *\/\n    transform: scale(1.02); \/* \u8f7b\u5fae\u653e\u5927\u6548\u679c *\/\n    box-shadow: 0 2px 4px rgba(0,0,0,0.1); \/* \u589e\u52a0\u8f7b\u5fae\u9634\u5f71 *\/\n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime = 0; document.querySelector('video').play();\">\n\u95ee\u9898\uff081\uff09\u6c42\\(f'(x)\\)\uff0c\u5e76\u5224\u65ad\u5355\u8c03\u533a\u95f4\uff1b\n<\/button>\n\n\n\n<br>\n\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left; \/* \u5de6\u5bf9\u9f50\u6587\u672c *\/\n   \n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime = 92; document.querySelector('video').play();\">\n    \u95ee\u9898\uff082\uff09\u7b2c\u4e00\u6b65\uff1a\u5206\u6790\u5f97\u51fa\\(f(\\displaystyle\\frac{1}{a}) = f(\\displaystyle\\frac{1}{b})\\);\n\n<\/button>\n\n\n\n<br>\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left; \/* \u5de6\u5bf9\u9f50\u6587\u672c *\/\n   \n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime = 192; document.querySelector('video').play();\">\n\n\u7b2c\u4e8c\u6b65\uff1a\u8bc1\u660e\\(\\displaystyle\\frac{1}{a} + \\displaystyle\\frac{1}{b} > 2\\);\n\n<\/button>\n\n\n\n<br>\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left; \/* \u5de6\u5bf9\u9f50\u6587\u672c *\/\n   \n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime = 497; document.querySelector('video').play();\">\n\n\u7b2c\u4e09\u6b65\uff1a\u8bc1\u660e\\(\\displaystyle\\frac{1}{a} + \\displaystyle\\frac{1}{b} < e\\).\n\n<\/button>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848\u89e3\u6790<\/summary>\n \n\n\n\n<p>\uff081\uff09\u7531\u9898\u610f\u5f97\\( f(x) = x(1 &#8211; \\ln x) \\)\uff0c\u5b9a\u4e49\u57df\\( x \\in (1, +\\infty) \\)<\/p>\n<p>\\( f'(x) = 1 &#8211; (\\ln x + 1) = -\\ln x \\)<\/p>\n<p>\u4ee4\\( f'(x) > 0 \\) \u5373\\( -\\ln x > 0 \\) \uff0c\u5f97\\( x \\in (0,1) \\)<\/p>\n<p>\\( \\therefore f(x) \\)\u5728\\( (0,1] \\)\u4e0a\u5355\u8c03\u9012\u589e\uff0c\u5728\\( (1, +\\infty) \\)\u4e0a\u5355\u8c03\u9012\u51cf\u3002<\/p>\n\n<p>\uff082\uff09\u7531\\( b\\ln a &#8211; a\\ln b = a &#8211; b \\)\u5f97<\/p>\n<p>\\( \\frac{1}{a}\\left(1 &#8211; \\ln \\frac{1}{a}\\right) = \\frac{1}{b}\\left(1 &#8211; \\ln \\frac{1}{b}\\right) \\)\uff0c \u5373\\( f\\left(\\frac{1}{a}\\right) = f\\left(\\frac{1}{b}\\right) \\)<\/p>\n<p>\u2460\u8bc1\u660e\\(\\dfrac{1}{a} + \\dfrac{1}{b} > 2\\)\uff08\u6781\u503c\u70b9\u504f\u79fb\u6cd5\uff09<\/p>\n<p>\u8bbe\\(x_1 = \\dfrac{1}{a}\\)\uff0c\\(x_2 = \\dfrac{1}{b}\\) \u4e14 \\(0 < x_1 < 1 < x_2 < e\\)<\/p>\n<p>\u5373\u8bc1\\(x_1 + x_2 > 2\\)\uff0c\u5373\u8bc1\\(x_2 > 2 &#8211; x_1\\)<\/p>\n<p>\\(\\because x_2 \\in (1, e)\\) \uff0c\\(\\therefore 2 &#8211; x_1 \\in (1, 2)\\)\uff0c\\(\\therefore\\)\u53ea\u9700\u8bc1\\(f(x_2) < f(2 - x_1)\\)<\/p>\n<p>\\(\\because f(x_1) = f(x_2)\\)\uff0c\\(\\therefore\\)\u53ea\u9700\u8bc1\\(f(x_1) < f(2 - x_1)\\)<\/p>\n<p>\u8bbe\\(F(x) = f(x) &#8211; f(2 &#8211; x)\\)<\/p>\n<p>\\(\\therefore\\)\u53ea\u9700\u8bc1\\(F(x) = f(x) &#8211; f(2 &#8211; x) < 0\\) \u5728\u5b9a\u4e49\u57df\\((0, 1)\\)\u4e0a\u6052\u6210\u7acb<\/p>\n<p>\\(F'(x) = f'(x) + f'(2 &#8211; x) = -\\ln x &#8211; \\ln(2 &#8211; x)\\)<\/p>\n<p>\\(F&#8221;(x) = -\\dfrac{1}{x} + \\dfrac{1}{2 &#8211; x} = \\dfrac{2x &#8211; 2}{x(2 &#8211; x)}\\)<\/p>\n<p>\u7531\\(x\\)\u7684\u53d6\u503c\u8303\u56f4\u53ef\u77e5\uff0c\\(F&#8221;(x) < 0\\) \uff0c\\(\\therefore F'(x)\\)\u5728\\((0, 1)\\)\u5355\u8c03\u9012\u51cf\uff0c\\(F'(1) = 0\\)\uff0c\\(F'(x) > F'(1) = 0\\)<\/p>\n<p>\\(\\therefore F'(x)\\)\u5728\\((0, 1)\\)\u5355\u8c03\u9012\u589e\u4e14\\(F(1) = 0\\) \uff0c\\(\\therefore F(x) < F(1) = 0\\)<\/p>\n<p>\\(\\therefore f(x) &#8211; f(2 &#8211; x)\\) \u5728\u5b9a\u4e49\u57df\\((0, 1)\\)\u4e0a\u6052\u6210\u7acb<\/p>\n<p>\u6240\u4ee5\uff0c\\(\\dfrac{1}{a} + \\dfrac{1}{b} > 2\\)\u6210\u7acb<\/p>\n<p>\u2461\u8bc1\u660e\\( \\dfrac{1}{a} + \\dfrac{1}{b} < e \\)\uff08\u653e\u7f29\u6cd5\uff09<\/p>\n<p>\u5373\u8bc1\\( x_1 + x_2 < e \\)<\/p>\n<p>\\( \\because f(x_1) = f(x_2) \\)\uff0c\\( \\because x_1 \\in (0,1) \\)\uff0c\\( \\therefore \\ln x_1 < 0 \\)<\/p>\n<p>\\( \\therefore x_1(1 &#8211; \\ln x_1) = x_2(1 &#8211; \\ln x_2) > x_1 \\)<\/p>\n<p>\\( \\therefore x_2(1 &#8211; \\ln x_2) > x_1 \\)\uff0c\\( \\therefore x_2(2 &#8211; \\ln x_2) > x_1 + x_2 \\)<\/p>\n\n<p>\u8bbe\\( g(x) = 2x &#8211; \\ln x \\)\uff0c\u5b9a\u4e49\u57df\\( (1, e) \\)<\/p>\n<p>\\( g'(x) = 1 &#8211; \\ln x \\)\uff0c\\( \\because x \\in (1, e) \\)\uff0c\\( \\therefore g'(x) > 0 \\)\uff0c\\( g(x) \\)\u5728\u5b9a\u4e49\u57df\u4e0a\u5355\u8c03\u9012\u589e<\/p>\n<p>\\( \\therefore g(x) < g(e) = e \\)\uff0c\u5373\\( x_1 + x_2 < e \\)\uff0c\u5373\\( \\dfrac{1}{a} + \\dfrac{1}{b} < e \\)<\/p>\n<\/details>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u9898\u75312023\u7ea725\u73ed\u6210\u5141\u822a\u8bb2\u89e3\uff01 [ratemypost] \u9898\u76ee\u5c55\u793a\u2014\u2014\u5148\u63a2\u7a76\u3001\u518d\u542c\u8bb2 \u9898\u76ee3 \u5df2\u77e5\u51fd\u6570\\( [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-4370","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4370","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4370"}],"version-history":[{"count":21,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4370\/revisions"}],"predecessor-version":[{"id":4795,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4370\/revisions\/4795"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4370"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4370"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}