{"id":4667,"date":"2025-11-09T15:24:59","date_gmt":"2025-11-09T07:24:59","guid":{"rendered":"http:\/\/jhwk.online\/?p=4667"},"modified":"2025-11-09T17:24:30","modified_gmt":"2025-11-09T09:24:30","slug":"%e5%87%bd%e6%95%b0%e5%af%bc%e6%95%b03%e8%ae%b2%e8%a7%a3-3-2-2-2-2-2","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=4667","title":{"rendered":"\u51fd\u6570\u5bfc\u65705"},"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\u97e9\u4e30\u7fbd\u8bb2\u89e3\uff01<\/p><\/div>\n\n\n\n<figure class=\"wp-block-video\"><video height=\"734\" style=\"aspect-ratio: 1306 \/ 734;\" width=\"1306\" controls src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/11\/\u51fd\u6570\u5bfc\u65705.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\u8bbe\u51fd\u6570\\( f(x)=x^{2}(e^{x}-a) \\)<p>\n(1)\u5f53\\( a=0 \\)\u65f6,\u6c42\u66f2\u7ebf\\( y=f(x) \\)\u5728\\( (1,f(1)) \\)\u5904\u7684\u5207\u7ebf\u65b9\u7a0b.<p>\n(2)\u82e5\\( f(x) \\)\u662f\u589e\u51fd\u6570,\u6c42\\( a \\)\u7684\u503c.<p>\n(3)\u5f53\\( 0\\lt a\\lt 1 \\)\u65f6,\u8bbe\\( x_{0} \\)\u4e3a\\( f(x) \\)\u7684\u6781\u5927\u503c\u70b9,\u8bc1\u660e\\( 0\\lt f(x_{0})\\lt \\displaystyle\\frac{4}{e} \\).<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-58b20738b8080c684d5fff23cf040d12\" 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(1) \u5207\u7ebf\u65b9\u7a0b\u4e3a$3ex-y-2e=0;$\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 =70; document.querySelector('video').play();\">\n\n\u95ee\u9898(2) \u89e3\u5f97$a=1$\uff1b\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 =290; document.querySelector('video').play();\">\n\n\u95ee\u9898(3) \u8bc1\u660e\u3002\n\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\u89e3:\n(1)\\( a=0 \\)\u65f6 \\( f(x)=x^{2}e^{x}, f'(x)=(x^{2})&#8217;e^{x}+x^{2}(e^{x})&#8217;=(x^{2}+2x)e^{x} \\)<p>\n\\( f(1)=e \\) \\( f'(1)=3e \\)<p>\n\\(\\therefore y=f(x)\\)\u5728\\( (1,f(1)) \\)\u5904\u5207\u7ebf\u4e3a \\( y-f(1)=f'(1)(x-1) \\)\n\u5373 \\( 3ex-y-2e=0 \\)<p>\n(2)\\( f'(x)= [x^{2}(e^{x}-a)]&#8217; \\)\n\\(=(x^{2})'(e^{x}-a)+(x^{2})(e^{x}-a)&#8217; \\)<p>\n\\(= 2x(e^{x}-a)+x^{2}e^{x} \\)\n\\(=x(2e^{x}+xe^{x}-2a) \\)<p>\n\u8bbe\\( g(x)=2e^{x}+xe^{x}-2a=(2+x)e^{x}-2a \\)<p>\n\\( g'(x)=2e^{x}+xe^{x}+x(e^{x})&#8217; \\)\n\\(=(3+x)e^{x} \\)<p>\n\u4ee4\\(\\begin{cases} g'(x)>0 \\\\ x\\in R \\end{cases}\\)\n\u89e3\u5f97 \\( x>-3 \\)\n\\(\\therefore g(x) \\uparrow (-3,+\\infty) \\)\n\\(\\downarrow (-\\infty,-3) \\)\n\n<p>\\( f(x) = x^2(e^x &#8211; a), f'(x) = x(2e^x + xe^x &#8211; 2a) = x[(2 + x)e^x &#8211; 2a] \\geq 0 \\)<\/p>\n<p>\u2460 \\( x \\geq 0 \\)\u65f6\uff0c<\/p>\n<p>\\( (2 + x)e^x &#8211; 2a \\geq 0, \\)\u5373\\( 2a \\leq (2 + x)e^x \\)<\/p>\n<p>\u8bbe\\( B(x) = (2 + x)e^x, \\uparrow [0, +\\infty) \\)<\/p>\n<p>\\( B(x) \\geq B(0) = 2, \\therefore 2a \\leq 2, \\ a \\leq 1 \\)<\/p>\n<p>\u2461 \\( x < 0 \\)\u65f6<\/p>\n<p>\\( (2 + x)e^x &#8211; 2a \\leq 0, \\)\u5373\\( 2a \\geq (2 + x)e^x \\)<\/p>\n<p>\\( B(x) = (2 + x)e^x (x < 0) \\)<\/p>\n<p>\\( B'(x) = (2 + x)e^x + (2 + x)(e^x)&#8217; = (3 + x)e^x \\)<\/p>\n<p>\u4ee4\\( \\begin{cases} B'(x) = (3 + x)e^x > 0 \\\\ x < 0 \\end{cases} \\)<\/p>\n<p>\u89e3\u5f97\\( x \\in (-3, 0] \\)<\/p>\n<p>\\( f(x) = x^2(e^x &#8211; a), f'(x) = x(2e^x + xe^x &#8211; 2a) = x[(2 + x)e^x &#8211; 2a] \\geq 0 \\)<\/p>\n<p>\u2460 \\( x \\geq 0 \\)\u65f6\uff0c<\/p>\n<p>\\( (2 + x)e^x &#8211; 2a \\geq 0, \\)\u5373\\( 2a \\leq (2 + x)e^x \\)<\/p>\n<p>\u8bbe\\( B(x) = (2 + x)e^x, \\uparrow [0, +\\infty) \\)<\/p>\n<p>\\( B(x) \\geq B(0) = 2, \\therefore 2a \\leq 2, \\ a \\leq 1 \\)<\/p>\n<p>\u2461 \\( x < 0 \\)\u65f6<\/p>\n<p>\\( (2 + x)e^x &#8211; 2a \\leq 0, \\)\u5373\\( 2a \\geq (2 + x)e^x \\)<\/p>\n<p>\\( B(x) = (2 + x)e^x (x < 0) \\)<\/p>\n<p>\\( B'(x) = (2 + x)e^x + (2 + x)(e^x)&#8217; = (3 + x)e^x \\)<\/p>\n<p>\u4ee4\\( \\begin{cases} B'(x) = (3 + x)e^x > 0 \\\\ x < 0 \\end{cases} \\)<\/p>\n<p>\u89e3\u5f97\\( x \\in (-3, 0] \\)<\/p>\n<p>\\( \\therefore B(x), \\uparrow [-3, 0], \\downarrow (-\\infty, -3) \\)<\/p>\n<p>\\( x \\to -\\infty \\ \\ B(x) \\to 0 \\)<\/p>\n<p>\\( x = 0 \\ \\ B(0) = 2 \\)<\/p>\n<p>\\( \\therefore B(x) (x \\leq 0) \\)\u56fe\u8c61\u4e3a<\/p>\n<p><img decoding=\"async\" src=\"B(x)\u56fe\u8c61\" alt=\"B(x)\u56fe\u8c61\"><\/p>\n<p>\\( \\therefore 2a \\geq 2, a \\geq 1 \\)<\/p>\n<p>\u7531\u2460\u2461\u5f97 \\( a = 1 \\)<\/p>\n<p>\uff083\uff09\u5f53\\(0 < a < 1\\)\u65f6\uff0c\u8bbe\\(x_0\\)\u4e3a\\(f(x)\\)\u7684\u6781\u5927\u503c<\/p>\n<p>\u8bc1\u660e\\(0 < f(x_0) < \\displaystyle \\frac{4}{e^2}\\)<\/p>\n<p>\u89e3\uff1a\\(f'(x) = xg(x)\\) \\(g(x) = 2e^x + xe^x &#8211; 2a\\)<\/p>\n<p>\\(g(0) = 2 &#8211; 2a > 0\\)<\/p>\n<p>\\(g(-2) = -2a < 0\\)<\/p>\n<p>\\(\\because g(x)\\)\u5728\\((-3, +\\infty)\\)\u4e2a<\/p>\n<p>\\(\\therefore\\)\u5b58\u5728\u552f\u4e00\\(x_0 \\in (-2, 0)\\)<\/p>\n<p>\u4f7f\\(g(x_0) = (x_0 + 2)e^{x_0} &#8211; 2a = 0\\)<\/p>\n<p>\\(\\therefore a = \\displaystyle \\frac{e^{x_0}(x_0 + 2)}{2}\\)<\/p>\n<p>\\(f(x_0) = x_0^2 (e^{x_0} &#8211; a)\\)<\/p>\n<p>\\(= x_0^2 \\left[e^{x_0} &#8211; \\displaystyle \\frac{e^{x_0}(x_0 + 2)}{2}\\right] = -\\displaystyle \\frac{x_0^3 e^{x_0}}{2}\\)<\/p>\n<p>\u8bbe\\(h(x) = \\displaystyle \\frac{x^3 e^x}{2}\\) \\(x \\in (-2, 0)\\) \\(h'(x)\\)<\/p>\n<p>\\(h'(x) = -\\displaystyle \\frac{(3x^2)e^x + x^3(e^x)}{2} = -\\displaystyle \\frac{(x^2 + 3x^2)e^x}{2} = -\\displaystyle \\frac{x^2(x + 3)e^x}{2} < 0\\)<\/p>\n<p>\\(\\therefore h(x)\\)\u5728\\((-2, 0)\\)\u4e0a\u2193<\/p>\n<p>\\(\\therefore h(0) < h(x) < h(-2)\\)<\/p>\n<p>\\(\\therefore h(0) < f(x_0) < h(-2)\\)<\/p>\n<p>\\(0 < f(x_0) < \\displaystyle \\frac{4}{e^2}\\)<\/p>\n<p>\u5f97\u8bc1<\/p>\n<\/details>\n\n\n\n \n","protected":false},"excerpt":{"rendered":"<p>\u672c\u9898\u75312023\u7ea725\u73ed\u97e9\u4e30\u7fbd\u8bb2\u89e3\uff01 [ratemypost] \u9898\u76ee\u5c55\u793a\u2014\u2014\u5148\u63a2\u7a76\u3001\u518d\u542c\u8bb2 \u8bbe\u51fd\u6570\\( f(x) [&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-4667","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\/4667","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=4667"}],"version-history":[{"count":11,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4667\/revisions"}],"predecessor-version":[{"id":4685,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4667\/revisions\/4685"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4667"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4667"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4667"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}