{"id":4559,"date":"2025-11-02T22:56:44","date_gmt":"2025-11-02T14:56:44","guid":{"rendered":"http:\/\/jhwk.online\/?p=4559"},"modified":"2025-11-04T17:01:09","modified_gmt":"2025-11-04T09:01:09","slug":"%e5%87%bd%e6%95%b0%e5%af%bc%e6%95%b03%e8%ae%b2%e8%a7%a3-3-2","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=4559","title":{"rendered":"\u51fd\u6570\u5bfc\u6570\u9898\u76ee1"},"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\u71d5\u5b50\u6052\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 src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/11\/\u51fd\u6570\u5bfc\u65701-1.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\u76ee1<\/h5>\n<p>\u5df2\u77e5\u51fd\u6570\\( f(x) = x^2 &#8211; 2a\\ln x + 1 \\)\uff0c\\( a \\in R \\)\u3002<p>\n<p>\uff081\uff09\u5f53\\( a = -1 \\)\u65f6\uff0c\u8bbe\u66f2\u7ebf\\( y = f(x) \\)\u5728\\( x = 1 \\)\u5904\u7684\u5207\u7ebf\u4e3a\\( l \\)\uff0c\u6c42\\( l \\)\u4e0e\u66f2\u7ebf\\( y = f(x) \\)\u7684\u516c\u5171\u70b9\u4e2a\u6570\uff1b<\/p>\n<p>\uff082\uff09\u5f53\\( a > 0 \\)\u65f6\uff0c\u82e5\\( \\forall x_1, x_2 \\in [1, e] \\)\uff0c\\( |f(x_1) &#8211; f(x_2)| < e^2 + 1 \\)\u6052\u6210\u7acb\uff0c\u6c42\u5b9e\u6570\\( a \\)\u7684\u53d6\u503c\u8303\u56f4\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\u7b2c\u4e00\u6b65\uff1a\u6c42\u51fa\u5207\u7ebf\u65b9\u7a0b\u4e3a\\(y=4x-2\\)\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 = 80; document.querySelector('video').play();\">\n    \u95ee\u9898\uff081\uff09\u7b2c\u4e8c\u6b65\uff1a\u8054\u7acb\u5207\u7ebf\u4e0e\\(y=f(x)\\),\u6c42\u89e3\u7684\u4e2a\u6570\u3002\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 =205; document.querySelector('video').play();\">\n\n\u95ee\u9898\uff082\uff09\u7b2c\u4e00\u6b65\uff1a\u5728\u7ed9\u5b9a\u8303\u56f4\u5185\u6c42\u5bfc\u51fd\u6570\u3002\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 =277; document.querySelector('video').play();\">\n\n\u95ee\u9898\uff082\uff09\u5206\u56db\u79cd\u60c5\u51b5\u8ba8\u8bba\\(a\\)\u3002\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>\u89e3: (1)\u5f53\\( a = -1 \\)\u65f6,\\( f(x) = x^2 + 2\\ln x + 1 \\),<p>\r\n<p>\u5219\\( f'(x) = 2x + \\displaystyle \\frac{2}{x} \\),\u6240\u4ee5\\( f'(1) = 4 \\),<\/p>\r\n<p>\u53c8\\( f(1) = 2 \\),\u6240\u4ee5\u66f2\u7ebf\\( y = f(x) \\)\u5728\\( x = 1 \\)\u5904\u7684\u5207\u7ebf\u65b9\u7a0b\u4e3a\\( y = 4(x &#8211; 1) + 2 \\),\u5373\\( y = 4x &#8211; 2 \\),<\/p>\r\n<p>\u8054\u7acb\u65b9\u7a0b\\( \\begin{cases} y = 4x &#8211; 2 \\\\ y = x^2 + 2\\ln x + 1 \\end{cases} \\)\uff0c\u53ef\u5f97\\( x^2 + 2\\ln x &#8211; 4x + 3 = 0 \\),<\/p>\r\n<p>\u4ee4\\( g(x) = x^2 + 2\\ln x &#8211; 4x + 3, x > 0 \\),<\/p>\r\n<p>\u5219\\( g'(x) = 2x + \\displaystyle \\frac{2}{x} &#8211; 4 \\geq 0 \\),<\/p>\r\n<p>\u6240\u4ee5\\( g(x) \\)\u5728\\( (0, +\\infty) \\)\u4e0a\u5355\u8c03\u9012\u589e.<\/p>\r\n<p>\u53c8\u56e0\u4e3a\\( g(1) = 0 \\),\u6240\u4ee5\\( g(x) = 0 \\)\u6709\u4e14\u4ec5\u67091\u89e3,<\/p>\r\n<p>\u6240\u4ee5\u76f4\u7ebf\\( l \\)\u4e0e\u66f2\u7ebf\\( y = f(x) \\)\u516c\u5171\u70b9\u4e2a\u6570\u4e3a1\uff1b<\/p>\r\n<p>(2)\\( \\forall x_1, x_2 \\in [1, e], |f(x_1) &#8211; f(x_2)| < e^2 + 1 \\)\u6052\u6210\u7acb,<\/p>\r\n<p>\u7b49\u4ef7\u4e8e\\( f(x)_{\\max} &#8211; f(x)_{\\min} < e^2 + 1 \\),<\/p>\r\n<p>\u7531\\( f(x) = x^2 &#8211; 2a\\ln x + 1 \\),\u5f97\\( f'(x) = 2x &#8211; \\displaystyle \\frac{2a}{x} = \\displaystyle \\frac{2(x^2 &#8211; a)}{x}, a > 0, x > 0 \\),<\/p>\r\n<p>\u5f53\\( x \\in (0, \\sqrt{a}) \\)\u65f6,\\( f'(x) < 0 \\),\u6240\u4ee5\\( f(x) \\)\u5728\\( (0, \\sqrt{a}) \\)\u4e0a\u5355\u8c03\u9012\u51cf\uff1b<\/p>\r\n<p>\u5f53\\( x \\in (\\sqrt{a}, +\\infty) \\)\u65f6,\\( f'(x) > 0 \\),\u6240\u4ee5\\( f(x) \\)\u5728\\( (\\sqrt{a}, +\\infty) \\)\u4e0a\u5355\u8c03\u9012\u589e,<\/p>\r\n<p>\u2460\u5f53\\( \\sqrt{a} \\leq 1 \\),\u5373\\( 0 < a \\leq 1 \\)\u65f6,\\( f(x) \\)\u5728\\( [1, e] \\)\u4e0a\u5355\u8c03\u9012\u589e.<\/p>\r\n<p>\u6240\u4ee5\\( f(x)_{\\min} = f(1) = 2, f(x)_{\\max} = f(e) = e^2 &#8211; 2a + 1 \\),\u6b64\u65f6\\( e^2 &#8211; 2a + 1 &#8211; 2 = e^2 &#8211; 2a &#8211; 1 < e^2 + 1 \\),\u6240\u4ee5\\( 0 < a \\leq 1 \\)\u6ee1\u8db3\u6761\u4ef6\uff1b<\/p>\r\n<p>\u2461\u5f53\\( \\sqrt{a} \\geq e \\),\u5373\\( a \\geq e^2 \\)\u65f6,\\( f(x) \\)\u5728\\( [1, e] \\)\u4e0a\u5355\u8c03\u9012\u51cf, \u6240\u4ee5\\( f(x)_{\\max} = f(1) = 2, f(x)_{\\min} = f(e) = e^2 &#8211; 2a + 1 \\), \u6b64\u65f6\\( 2 &#8211; e^2 + 2a &#8211; 1 = 2a &#8211; e^2 + 1 \\geq e^2 + 1 \\),<\/p>\r\n<p>\u4e0d\u5408\u9898\u610f,<\/p>\r\n<p>\u2462\u5f53\\( 1 < \\sqrt{a} < e \\),\u5373\\( 1 < a < e^2 \\)\u65f6,\\( f(x) \\)\u5728\\( (1, \\sqrt{a}) \\)\u4e0a\u5355\u8c03\u9012\u51cf,\u5728\\( (\\sqrt{a}, e) \\)\u4e0a\u5355\u8c03\u9012\u589e,<\/p>\r\n<p>\u6240\u4ee5\\( f(x)_{\\min} = f(\\sqrt{a}) = a &#8211; a\\ln a + 1, f(x)_{\\max} = \\max\\{f(1), f(e)\\} \\),<\/p>\r\n<p>\u6240\u4ee5\\( 2 &#8211; (a &#8211; a\\ln a + 1) < e^2 + 1 \\),\u4e14\\( (e^2 - 2a + 1) - (a - a\\ln a + 1) < e^2 + 1 \\),\u5373\\( a\\ln a - a - e^2 < 0 \\),\u4e14\\( a\\ln a - 3a - 1 < 0 \\),<\/p>\r\n<p>\u4ee4\\( m(a) = a\\ln a &#8211; a &#8211; e^2 \\),\u5f53\\( 1 < a < e^2 \\)\u65f6,\\( m'(a) = \\ln a > 0 \\),\u6240\u4ee5\\( m(a) \\)\u5728\\( (1, e^2) \\)\u4e0a\u5355\u8c03\u9012\u589e,\u6240\u4ee5\\( m(a) < m(e^2) = 0 \\),<\/p>\r\n<p>\u4ee4\\( t(a) = a\\ln a &#8211; 3a &#8211; 1 \\),\u5f53\\( 1 < a < e^2 \\)\u65f6,\\( t'(a) = \\ln a - 2 < 0 \\), \u6240\u4ee5\\( t(a) \\)\u5728\\( (1, e^2) \\)\u4e0a\u5355\u8c03\u9012\u51cf,\u6240\u4ee5\\( t(a) < t(1) = -4 < 0 \\),<\/p>\r\n<p>\u6240\u4ee5\u5f53\\( 1 < a < e^2 \\)\u65f6,\u6ee1\u8db3\u9898\u610f.<\/p>\r\n<p>\u7efc\u4e0a, \\( a \\)\u7684\u53d6\u503c\u8303\u56f4\u4e3a\\( a \\in (0, e^2) \\)\u3002<\/p>\n<\/details>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u9898\u75312023\u7ea725\u73ed\u71d5\u5b50\u6052\u8bb2\u89e3\uff01 [ratemypost] \u9898\u76ee\u5c55\u793a\u2014\u2014\u5148\u63a2\u7a76\u3001\u518d\u542c\u8bb2 \u9898\u76ee1 \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-4559","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\/4559","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=4559"}],"version-history":[{"count":6,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4559\/revisions"}],"predecessor-version":[{"id":4595,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4559\/revisions\/4595"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4559"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4559"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}