{"id":4448,"date":"2025-10-27T11:21:14","date_gmt":"2025-10-27T03:21:14","guid":{"rendered":"http:\/\/47.95.222.19\/?p=4448"},"modified":"2025-11-15T23:05:45","modified_gmt":"2025-11-15T15:05:45","slug":"%e5%87%bd%e6%95%b0%e5%af%bc%e6%95%b03%e8%ae%b2%e8%a7%a3-2","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=4448","title":{"rendered":"\u51fd\u6570\u5bfc\u65704\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\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/11\/\u51fd\u6570\u5bfc\u65703.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\u76ee4<\/h5>\n<p>\u5df2\u77e5\u51fd\u6570\\( f(x)= e^{x\\cos x} \\)\uff08\\( e \\)\u4e3a\u81ea\u7136\u5bf9\u6570\u7684\u5e95\u6570\uff09<\/p>\n\n<p>\uff081\uff09\u6c42\u51fd\u6570\\( f(x) \\)\u5728\u70b9\\((0,f(0))\\)\u5904\u7684\u5207\u7ebf\u65b9\u7a0b\uff1b<\/p>\n\n<p>\uff082\uff09\u82e5\u8bb0\\( g(x)=-x^{3}-3x\\cdot \\cos x+(a+3)x \\)\uff0c\u82e5\\(\\forall x\\in [0,1] \\)\uff0c\u6709\\( f(x)\\leq e^{g(x)} \\)\uff0c\u6c42\\( a \\)\u7684\u53d6\u503c\u8303\u56f4\uff1b<\/p>\n\n<p>\uff083\uff09\u8bbe\\( n\\in N^{*} \\)\uff0c\u4e14\\( n\\geq 2 \\)\uff0c\u8bc1\u660e\uff1a\\( \\cos1+\\cos\\displaystyle\\frac{2}{3}\\)+\\(\\cos\\displaystyle\\frac{1}{2}\\)+\\(\\cos\\displaystyle\\frac{2}{5}\\)+\\(\\dots+\\cos\\displaystyle\\frac{2}{n} < n-\\displaystyle\\frac{3}{2}\\)+\\(\\displaystyle\\frac{1}{n+1} \\)\uff0e<\/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\u7b2c\u4e00\u95ee\uff0c\u5229\u7528\u5207\u7ebf\u6c42<span data-type=\"inline-math\" data-value=\"IGYoeCkg\"><\/span>\u659c\u7387\u7ed3\u5408\u70b9\u659c\u5f0f\u6c42\u89e3\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 = 154; document.querySelector('video').play();\">\n    \u7b2c\u4e8c\u95ee\uff0c\u5206\u6790\u9898\u76ee\uff0c\u5229\u7528\u5206\u53c2\u6cd5\u6c42\u89e3<span data-type=\"inline-math\" data-value=\"IGEg\"><\/span>\u7684\u53d6\u503c\u8303\u56f4;\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 = 511; document.querySelector('video').play();\">\n\n\u7b2c\u4e09\u95ee\uff0c\u5206\u6790\u9898\u76ee\uff0c\u5229\u7528\u7b2c\u4e8c\u95ee\u7ed3\u8bba\u548c\u653e\u7f29\u6cd5\u5904\u7406<span data-type=\"inline-math\" data-value=\"XGZyYWN7MX17bl4yfQ==\"><\/span>\u6c42\u503c\uff0c\u7ed3\u5408\u7d2f\u52a0\u6cd5\u5b8c\u6210;\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 = 859; document.querySelector('video').play();\">\n\n\u7b2c\u56db\u95ee\uff0c\u603b\u7ed3\u9898\u76ee\uff0c\u603b\u4f53\u5206\u6790.\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\n\u51fd\u6570 \\( f(x) = e^{x\\sin x} \\)\uff0c\u5b9a\u4e49\u57df\u4e3a \\( (-\\infty, +\\infty) \\)\u3002<\/p>\n<p>\u6c42\u5bfc\u5f97 \\( f'(x) = e^{x\\cos x} \\cdot (\\cos x &#8211; x\\sin x) \\)\u3002<\/p>\n<p>\u5f53 \\( x = 0 \\) \u65f6\uff0c\\( f'(0) = 1 \\)\uff0c\\( f(0) = 1 \\)\uff0c<\/p>\n<p>\u6240\u4ee5 \\( f(x) \\) \u5728 \\( (0,1) \\) \u5904\u7684\u5207\u7ebf\u65b9\u7a0b\u4e3a \\( y &#8211; 1 = x \\)\uff0c\u5373 \\( x &#8211; y + 1 = 0 \\)\u3002<\/p>\n\n<p>\uff082\uff09\n\u7531 \\( f(x) \\leq e^{g(x)} \\)\uff0c\u53ef\u5f97 \\( x\\cos x \\leq g(x) \\)\uff0c\u5373 \\( x\\cos x \\leq -x^3 &#8211; 3x\\cos x + (a + 3)x \\)\u3002<\/p>\n<p>&#8211; \u5f53 \\( x = 0 \\) \u65f6\uff0c\\( 0 \\leq 0 \\) \u6210\u7acb\uff0c\u6b64\u65f6 \\( a \\in \\mathbb{R} \\)\u3002<\/p>\n<p>&#8211; \u5f53 \\( x \\in (0,1] \\) \u65f6\uff0c\u5316\u7b80\u5f97 \\( -x^2 &#8211; 4\\cos x + a + 3 \\geq 0 \\)\uff0c\u6574\u7406\u5f97 \\( a \\geq x^2 + 4\\cos x &#8211; 3 \\)\u3002<\/p>\n<p>\u8bbe \\( h(x) = x^2 + 4\\cos x &#8211; 3 \\)\uff0c\u5219 \\( h'(x) = 2x &#8211; 4\\sin x \\)\uff0c\\( h&#8221;(x) = 2 &#8211; 4\\cos x \\)\u3002<\/p>\n<p>\u56e0\u4e3a \\( x \\leq 1 < \\displaystyle \\frac{\\pi}{3} (\\approx 1.04) \\)\uff0c\u6240\u4ee5 \\( h''(x) < 0 \\)\uff0c\\( h'(x) \\) \u5728 \\( (0,1] \\) \u4e0a\u5355\u8c03\u9012\u51cf\uff0c<\/p>\n<p>\u6545 \\( h'(x) < h'(0) = 0 \\)\uff0c\\( h(x) \\) \u5728 \\( (0,1] \\) \u4e0a\u5355\u8c03\u9012\u51cf\uff0c<\/p>\n<p>\u6240\u4ee5 \\( h(x) < h(0) = 1 \\)\uff0c\u56e0\u6b64 \\( a \\geq 1 \\)\uff0c\u5373 \\( a \\in [1, +\\infty) \\)\u3002<\/p>\n\n<p>\uff083\uff09\n\u56e0\u4e3a \\( n \\in N^* \\)\uff0c\u6240\u4ee5 \\( \\displaystyle \\frac{2}{n} > 0 \\)\uff1b\u53c8\u56e0\u4e3a \\( n \\geq 2 \\)\uff0c\u6240\u4ee5 \\( \\displaystyle \\frac{2}{n} \\leq 1 \\)\uff0c\u5373 \\( \\displaystyle \\frac{2}{n} \\in (0,1] \\)\u3002<\/p>\n<p>\u7531\uff082\uff09\u5f97 \\( x^2 + 4\\cos x &#8211; 3 \\leq a \\)\uff0c\u5f53 \\( a = 1 \\) \u65f6\uff0c\\( x^2 + 4\\cos x &#8211; 3 \\leq 1 \\)\uff0c\u6574\u7406\u5f97 \\( \\cos x \\leq 1 &#8211; \\displaystyle \\frac{x^2}{4} \\)\u3002<\/p>\n<p>\u8bbe \\( x = \\displaystyle \\frac{2}{n} \\)\uff0c\u4ee3\u5165\u5f97 \\( \\cos\\displaystyle \\frac{2}{n} \\leq 1 &#8211; \\displaystyle \\frac{1}{n^2} \\)\u3002<\/p>\n<p>\u56e0\u4e3a \\( \\displaystyle \\frac{1}{n^2} > \\displaystyle \\frac{1}{n} \\cdot \\displaystyle \\frac{1}{n + 1} \\)\uff0c\u6240\u4ee5 \\( -\\displaystyle \\frac{1}{n^2} < -\\displaystyle \\frac{1}{n} \\cdot \\displaystyle \\frac{1}{n + 1} \\)\uff0c<\/p>\n<p>\u56e0\u6b64 \\( 1 &#8211; \\displaystyle \\frac{1}{n^2} < 1 - \\displaystyle \\frac{1}{n} \\cdot \\displaystyle \\frac{1}{n + 1} \\)\uff0c\u6545 \\( \\cos\\displaystyle \\frac{2}{n} \\leq 1 - \\displaystyle \\frac{1}{n^2} < 1 - \\displaystyle \\frac{1}{n} \\cdot \\displaystyle \\frac{1}{n + 1} \\)\uff0c\u5373 \\( \\cos\\displaystyle \\frac{2}{n} \\leq 1 - \\left( \\displaystyle \\frac{1}{n} - \\displaystyle \\frac{1}{n + 1} \\right) \\)\u3002<\/p>\n<p>\u4e24\u4fa7\u5206\u522b\u7d2f\u52a0\u5f97\uff1a<\/p>\n\n\n<p>  $\\cos 1$ + $\\cos\\displaystyle \\frac{2}{3}$ + $\\cos\\displaystyle \\frac{1}{2}$ + $\\cos\\displaystyle \\frac{2}{5}$ + $\\dots$ +$ \\cos\\displaystyle \\frac{2}{n}$  $\\leq $  $(n &#8211; 1)$-$(\\displaystyle \\frac{1}{2}$ -$ \\displaystyle \\frac{1}{3}$ +$ \\displaystyle \\frac{1}{3}$ &#8211; $\\displaystyle \\frac{1}{4}$ +$ \\displaystyle \\frac{1}{4}$ &#8211; $\\displaystyle \\frac{1}{5}$ + $\\dots$ +$ \\displaystyle \\frac{1}{n}$ -$ \\displaystyle \\frac{1}{n + 1})$       <\/p>\n<p>  $\\cos 1$ + $\\cos\\displaystyle \\frac{2}{3}$ + $\\cos\\displaystyle \\frac{1}{2}$ + $\\cos\\displaystyle \\frac{2}{5}$ + $\\dots$ +$ \\cos\\displaystyle \\frac{2}{n}$ $ \\leq$ $(n &#8211; 1)$ &#8211; $ \\displaystyle \\frac{1}{2}$ -$ \\displaystyle \\frac{1}{n + 1}$  <\/p>\n<p>  $\\cos 1$ + $\\cos\\displaystyle \\frac{2}{3}$ + $\\cos\\displaystyle \\frac{1}{2}$ + $\\cos\\displaystyle \\frac{2}{5}$ + $\\dots$ +$ \\cos\\displaystyle \\frac{2}{n}$ $ \\leq$  $n$ &#8211; $\\displaystyle \\frac{3}{2}$ + $\\displaystyle \\frac{1}{n + 1}$  <\/p>\n<\/details>\n\n\n\n \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\u76ee4 \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-4448","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\/4448","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=4448"}],"version-history":[{"count":32,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4448\/revisions"}],"predecessor-version":[{"id":4796,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4448\/revisions\/4796"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4448"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4448"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}