{"id":10934,"date":"2023-08-20T16:15:01","date_gmt":"2023-08-20T08:15:01","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=10934"},"modified":"2025-08-17T13:52:26","modified_gmt":"2025-08-17T05:52:26","slug":"%e5%af%bc%e6%95%b0%e7%9a%84%e5%ba%94%e7%94%a8","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=10934","title":{"rendered":"\u5bfc\u6570\u7684\u5e94\u7528"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span 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href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E5%87%BD%E6%95%B0%E7%9A%84%E5%8D%95%E8%B0%83%E6%80%A7\" >\u51fd\u6570\u7684\u5355\u8c03\u6027<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E5%80%BC\" >\u51fd\u6570\u7684\u6781\u503c<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%9E%81%E5%80%BC%E7%9A%84%E5%BF%85%E7%84%B6%E6%9D%A1%E4%BB%B6\" >\u6781\u503c\u7684\u5fc5\u7136\u6761\u4ef6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%9E%81%E5%80%BC%E7%9A%84%E7%AC%AC%E4%B8%80%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\" >\u6781\u503c\u7684\u7b2c\u4e00\u5145\u5206\u6761\u4ef6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%9E%81%E5%80%BC%E7%9A%84%E7%AC%AC%E4%BA%8C%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\" >\u6781\u503c\u7684\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E5%87%BD%E6%95%B0%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC\" >\u51fd\u6570\u7684\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%87%B9%E5%87%B8%E6%80%A7\" >\u66f2\u7ebf\u7684\u51f9\u51f8\u6027<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E5%87%B9%E5%87%B8%E5%88%A4%E5%AE%9A\" >\u51f9\u51f8\u5224\u5b9a<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%8B%90%E7%82%B9\" >\u62d0\u70b9<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E5%BF%85%E8%A6%81%E6%9D%A1%E4%BB%B6\" >\u5fc5\u8981\u6761\u4ef6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E7%AC%AC%E4%B8%80%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\" >\u7b2c\u4e00\u5145\u5206\u6761\u4ef6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E7%AC%AC%E4%BA%8C%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\" >\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%9B%B2%E7%BA%BF%E7%9A%84%E6%B8%90%E8%BF%91%E7%BA%BF\" >\u66f2\u7ebf\u7684\u6e10\u8fd1\u7ebf<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%B0%B4%E5%B9%B3%E6%B8%90%E8%BF%91%E7%BA%BF\" >\u6c34\u5e73\u6e10\u8fd1\u7ebf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E9%93%85%E7%9B%B4%E6%B8%90%E8%BF%91%E7%BA%BF\" >\u94c5\u76f4\u6e10\u8fd1\u7ebf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%96%9C%E6%B8%90%E8%BF%91%E7%BA%BF\" >\u659c\u6e10\u8fd1\u7ebf<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E6%9B%B2%E7%BA%BF%E5%92%8C%E5%BC%A7%E5%BE%AE%E5%88%86%E5%92%8C%E6%9B%B2%E7%8E%87\" >\u66f2\u7ebf\u548c\u5f27\u5fae\u5206\u548c\u66f2\u7387<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/blog.iyatt.com\/?p=10934\/#%E7%9B%B8%E5%85%B3%E5%8F%98%E5%8C%96%E7%8E%87\" >\u76f8\u5173\u53d8\u5316\u7387<\/a><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"%E5%87%BD%E6%95%B0%E7%9A%84%E5%8D%95%E8%B0%83%E6%80%A7\"><\/span>\u51fd\u6570\u7684\u5355\u8c03\u6027<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8bbe<code>f(x)<\/code>\u5728<code>[a,b]<\/code>\u5185\u8fde\u7eed\uff0c\u5728<code>(a,b)<\/code>\u5185\u53ef\u5bfc\u3002<br \/>\n\u2460\u82e5\u5728<code>(a,b)<\/code>\u5185\uff0c<code class=\"katex-inline\">f'(x)\\gt0<\/code>\uff0c\u5219<code>f(x)<\/code>\u5728<code>[a,b]<\/code>\u5355\u8c03\u9012\u589e\uff1b<br \/>\n\u2461\u82e5\u5728<code>(a,b)<\/code>\u5185\uff0c<code class=\"katex-inline\">f'(x)\\lt0<\/code>\uff0c\u5219<code>f(x)<\/code>\u5728<code>[a,b]<\/code>\u5355\u8c03\u9012\u51cf\uff1b<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E5%80%BC\"><\/span>\u51fd\u6570\u7684\u6781\u503c<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8bbe<code>y=f(x)<\/code>\u5728<code class=\"katex-inline\">U(x_0,\\delta)<\/code>\u6709\u5b9a\u4e49\u3002\u5982\u679c\u5bf9\u4e8e\u8be5\u90bb\u57df\u5185\u4efb\u4f55<code>x<\/code>\uff0c\u6052\u6709<code class=\"katex-inline\">f(x)\\le f(x_0)<\/code>\u6216<code class=\"katex-inline\">f(x)\\ge f(x_0)<\/code>\uff0c\u5219\u79f0<code class=\"katex-inline\">x_0<\/code>\u4e3a<code>f(x)<\/code>\u7684\u4e00\u4e2a\u6781\u5927\u503c\u70b9\u6216\u6781\u5c0f\u503c\u70b9\uff0c\u79f0<code class=\"katex-inline\">f(x_0)<\/code>\u4e3a<strong>\u6781\u5927\u503c<\/strong>\u6216<strong>\u6781\u5c0f\u503c<\/strong>\uff0c\u90fd\u7edf\u79f0\u4e3a<strong>\u6781\u503c<\/strong>\uff0c\u5bf9\u5e94\u7684\u70b9\u7edf\u79f0\u4e3a<strong>\u6781\u503c\u70b9<\/strong>\u3002\u5bfc\u6570\u4e3a0\u7684\u70b9\u79f0\u4e3a\u51fd\u6570\u7684\u9a7b\u70b9\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%9E%81%E5%80%BC%E7%9A%84%E5%BF%85%E7%84%B6%E6%9D%A1%E4%BB%B6\"><\/span>\u6781\u503c\u7684\u5fc5\u7136\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbe<code>y=f(x)<\/code>\u5728\u70b9<code class=\"katex-inline\">x_0<\/code>\u5904\u53ef\u5bfc\uff0c\u5982\u679c<code class=\"katex-inline\">x_0<\/code>\u4e3af(x)\u7684\u6781\u503c\u70b9\uff0c\u5219<code class=\"katex-inline\">f'(x_0)=0<\/code>\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%9E%81%E5%80%BC%E7%9A%84%E7%AC%AC%E4%B8%80%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\"><\/span>\u6781\u503c\u7684\u7b2c\u4e00\u5145\u5206\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code>f(x)<\/code>\u5728<code class=\"katex-inline\">\\mathring{U}(x_0,\\delta)<\/code>\u5185\u53ef\u5bfc\uff0c\u4e14<code class=\"katex-inline\">f'(x_0)=0<\/code>\u6216<code class=\"katex-inline\">f'(x_0)<\/code>\u4e0d\u5b58\u5728\u4f46\u662f<code>f(x)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u8fde\u7eed\uff08\u6bd4\u5982<code>f(x)=|x|<\/code>\u5728<code>x=0<\/code>\u5904\u5bfc\u6570\u4e0d\u5b58\u5728\uff0c\u4f46\u662f\u8fde\u7eed\uff09\u3002<br \/>\n\u82e5<code class=\"katex-inline\">x\\lt x_0<\/code>\u65f6\uff0c<code class=\"katex-inline\">f'(x)\\ge0<\/code>\uff1b<code class=\"katex-inline\">x>x_0<\/code>\u65f6\uff0c<code class=\"katex-inline\">f'(x)\\le0<\/code>\uff0c\u5219<code>f(x)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u53d6\u6781\u5927\u503c\uff1b<br \/>\n\u82e5<code class=\"katex-inline\">x\\lt x_0<\/code>\u65f6\uff0c<code class=\"katex-inline\">f'(x)\\le0<\/code>\uff1b<code class=\"katex-inline\">x>x_0<\/code>\u65f6\uff0c<code class=\"katex-inline\">f'(x)\\ge0<\/code>\uff0c\u5219<code>f(x)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u53d6\u6781\u5927\u503c\uff1b<br \/>\n\u82e5<code>f(x)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u4e24\u4fa7\u4e0d\u53d8\u53f7\uff0c\u5219<code class=\"katex-inline\">x_0<\/code>\u4e0d\u662f<code>f(x)<\/code>\u7684\u6781\u503c\u70b9\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%9E%81%E5%80%BC%E7%9A%84%E7%AC%AC%E4%BA%8C%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\"><\/span>\u6781\u503c\u7684\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">f'(x_0)=0\uff0cf''(x_0)\\ne0<\/code>\u65f6\uff0c<br \/>\n<code class=\"katex-inline\">f''(x_0)<0<\/code>\uff0c\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u4e3a\u6781\u5927\u503c\u70b9\uff1b<br \/>\n<code class=\"katex-inline\">f''(x_0)>0<\/code>\uff0c\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u4e3a\u6781\u5c0f\u503c\u70b9\uff1b<br \/>\n<code class=\"katex-inline\">f''(x_0)=0<\/code>\uff0c\u4e0d\u80fd\u5224\u5b9a<code class=\"katex-inline\">x_0<\/code>\u662f\u5426\u4e3a\u6781\u503c\u70b9\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E5%87%BD%E6%95%B0%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC\"><\/span>\u51fd\u6570\u7684\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8bbe\u51fd\u6570<code>y=f(x)<\/code>\u5728<code>[a,b]<\/code>\u6709\u5b9a\u4e49\uff0c<code class=\"katex-inline\">x_0\\in[a,b]<\/code>\uff0c\u82e5\u5bf9\u4e8e<code class=\"katex-inline\">\\forall x\\in[a,b]<\/code>\uff0c\u6052\u6709<code class=\"katex-inline\">f(x)\\le f(x_0)<\/code>\u6216<code class=\"katex-inline\">f(x)\\ge f(x_0)<\/code>\uff0c\u5219\u79f0<code class=\"katex-inline\">f(x_0)\u4e3a\u51fd\u6570<\/code>f(x)<code><\/code>\u5728<code>[a,b]<\/code>\u4e0a\u7684<strong>\u6700\u5927\u503c<\/strong>\u6216<strong>\u6700\u5c0f\u503c<\/strong>\uff0c\u79f0<code class=\"katex-inline\">x_0<\/code>\u4e3a<code>f(x)<\/code>\u5728<code>[a,b]<\/code>\u4e0a\u7684<strong>\u6700\u5927\u503c\u70b9<\/strong>\u6216<strong>\u6700\u5c0f\u503c\u70b9<\/strong>\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%87%B9%E5%87%B8%E6%80%A7\"><\/span>\u66f2\u7ebf\u7684\u51f9\u51f8\u6027<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8bbe\u51fd\u6570<code>f(x)<\/code>\u5728\u533a\u95f4<code>I<\/code>\u4e0a\u8fde\u7eed\uff0c\u5982\u679c\u5bf9<code>I<\/code>\u4e0a\u4efb\u610f\u4e24\u70b9<code class=\"katex-inline\">x_1,x_2<\/code>\uff0c\u6052\u6709<\/p>\n<pre><code class=\"language-katex\">f(\\frac{x_1+x_2}{2})\\lt\\frac{f(x_1)+f(x_2)}{2}<\/code><\/pre>\n<p>\u5219\u79f0<code>f(x)<\/code>\u5728<code>I<\/code>\u4e0a\u7684\u56fe\u5f62\u662f\u51f9\u7684\uff0c\u5982\u679c\u6052\u6709<\/p>\n<pre><code class=\"language-katex\">f(\\frac{x_1+x_2}{2})\\gt\\frac{f(x_1)+f(x_2)}{2}<\/code><\/pre>\n<p>\u5219\u79f0<code>f(x)<\/code>\u5728<code>I<\/code>\u4e0a\u7684\u56fe\u5f62\u662f\u51f8\u7684\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E5%87%B9%E5%87%B8%E5%88%A4%E5%AE%9A\"><\/span>\u51f9\u51f8\u5224\u5b9a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbe\u51fd\u6570<code>y=f(x)<\/code>\u5728<code>[a,b]<\/code>\u4e0a\u8fde\u7eed\uff0c\u5728<code>(a,b)<\/code>\u5185\u4e8c\u9636\u53ef\u5bfc<br \/>\n\u82e5\u5728<code>(a,b)<\/code>\u5185\u6709<code class=\"katex-inline\">f''(x)\\gt0<\/code>\uff0c\u5219<code>f(x)<\/code>\u5728<code>[a,b]<\/code>\u5185\u56fe\u5f62\u662f\u51f9\u7684\uff1b<br \/>\n\u82e5\u5728<code>(a,b)<\/code>\u5185\u6709<code class=\"katex-inline\">f''(x)\\lt0<\/code>\uff0c\u5219<code>f(x)<\/code>\u5728<code>[a,b]<\/code>\u5185\u56fe\u5f62\u662f\u51f8\u7684\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%8B%90%E7%82%B9\"><\/span>\u62d0\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8fde\u7eed\u66f2\u7ebf\u5f27\u4e0a\u7684\u51f9\u4e0e\u51f8\u7684\u5206\u754c\u70b9\u79f0\u4e3a\u66f2\u7ebf\u5f27\u7684<strong>\u62d0\u70b9<\/strong>\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E5%BF%85%E8%A6%81%E6%9D%A1%E4%BB%B6\"><\/span>\u5fc5\u8981\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u8bbe<code>y=f(x)<\/code>\u5728\u70b9<code class=\"katex-inline\">x_0<\/code>\u5904\u4e8c\u9636\u53ef\u5bfc\uff0c\u4e14\u70b9<code class=\"katex-inline\">(x_0,f(x_0))<\/code>\u4e3a\u66f2\u7ebf<code>y=f(x)<\/code>\u7684\u62d0\u70b9\uff0c\u5219<code class=\"katex-inline\">f''(x_0)=0<\/code>\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E7%AC%AC%E4%B8%80%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\"><\/span>\u7b2c\u4e00\u5145\u5206\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u8bbe<code>y=f(x)<\/code>\u5728<code class=\"katex-inline\">\\mathring{U}(x_0,\\delta)<\/code>\u4e8c\u9636\u53ef\u5bfc\uff0c\u4e14<code class=\"katex-inline\">f''(x_0)=0<\/code>\uff08\u6216<code>f(x)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u8fde\u7eed\uff09\u3002<br \/>\n\u82e5<code class=\"katex-inline\">f''(x_0)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5de6\u53f3\u4e24\u4fa7\u5f02\u53f7\uff0c\u5219\u70b9<code class=\"katex-inline\">(x_0,f(x_0))<\/code>\u662f\u66f2\u7ebf<code>y=f(x)<\/code>\u7684\u62d0\u70b9\uff1b<br \/>\n\u82e5<code class=\"katex-inline\">f''(x_0)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5de6\u53f3\u4e24\u4fa7\u540c\u53f7\uff0c\u5219\u70b9<code class=\"katex-inline\">(x_0,f(x_0))<\/code>\u4e0d\u662f\u66f2\u7ebf<code>y=f(x)<\/code>\u7684\u62d0\u70b9\uff1b<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E7%AC%AC%E4%BA%8C%E5%85%85%E5%88%86%E6%9D%A1%E4%BB%B6\"><\/span>\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u8bbe<code>y=f(x)<\/code>\u5728<code class=\"katex-inline\">x_0<\/code>\u5904\u4e09\u9636\u53ef\u5bfc\uff0c\u4e14<code class=\"katex-inline\">f''(x_0)=0<\/code>\u3002<\/p>\n<p>\u82e5<code class=\"katex-inline\">f'''(x_0)\\ne0<\/code>\uff0c\u5219\u70b9<code class=\"katex-inline\">(x_0, f(x_0))<\/code>\u662f\u66f2\u7ebf<code>y=f(x)<\/code>\u7684\u62d0\u70b9\uff1b<br \/>\n\u82e5<code class=\"katex-inline\">f'''(x_0)=0<\/code>\uff0c\u5219\u8be5\u65b9\u6cd5\u4e0d\u80fd\u5224\u65ad\u62d0\u70b9\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E6%9B%B2%E7%BA%BF%E7%9A%84%E6%B8%90%E8%BF%91%E7%BA%BF\"><\/span>\u66f2\u7ebf\u7684\u6e10\u8fd1\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u82e5\u70b9<code>M<\/code>\u6cbf\u66f2\u7ebf<code>y=f(x)<\/code>\u65e0\u9650\u8fdc\u79bb\u539f\u70b9\u65f6\uff0c\u5b83\u4e0e\u67d0\u6761\u5b9a\u76f4\u7ebf<code>L<\/code>\u4e4b\u95f4\u7684\u8ddd\u79bb\u5c06\u8d8b\u8fd1\u4e8e\u96f6\uff0c\u5219\u79f0\u76f4\u7ebf<code>L<\/code>\u4e3a\u66f2\u7ebf<code>y=f(x)<\/code>\u7684\u4e00\u6761\u6e10\u8fd1\u7ebf\u3002\u82e5\u76f4\u7ebf<code>L<\/code>\u4e0e<code>x<\/code>\u8f74\u5e73\u884c\uff0c\u5219\u79f0L\u4e3a\u66f2\u7ebf<code>y=f(X)<\/code>\u7684<strong>\u6c34\u5e73\u6e10\u8fd1\u7ebf<\/strong>\uff1b\u82e5\u76f4\u7ebf<code>L<\/code>\u4e0e<code>x<\/code>\u8f74\u5782\u76f4\uff0c\u5219\u79f0<code>L<\/code>\u4e3a\u66f2\u7ebf<code>y=f(x)<\/code>\u7684\u94c5\u76f4\u6e10\u8fd1\u7ebf\uff1b\u6392\u9664\u8fd9\u4e24\u79cd\u60c5\u51b5\uff0c\u5c31\u5c5e\u4e8e\u659c\u6e10\u8fd1\u7ebf\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%B0%B4%E5%B9%B3%E6%B8%90%E8%BF%91%E7%BA%BF\"><\/span>\u6c34\u5e73\u6e10\u8fd1\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">\\lim_{x\\to\\infty}f(x)=A<\/code>\uff0c<code>y=A<\/code>\u5c31\u662f<code>y=f(x)<\/code>\u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E9%93%85%E7%9B%B4%E6%B8%90%E8%BF%91%E7%BA%BF\"><\/span>\u94c5\u76f4\u6e10\u8fd1\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">\\lim_{x\\to x_0}f(x)=\\infty<\/code>\uff0c<code class=\"katex-inline\">x=x_0<\/code>\u5c31\u662f<code>y=f(x)<\/code>\u94c5\u76f4\u6e10\u8fd1\u7ebf<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%9C%E6%B8%90%E8%BF%91%E7%BA%BF\"><\/span>\u659c\u6e10\u8fd1\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u82e5<code class=\"katex-inline\">\\lim_{x\\to\\infty}\\frac{f(x)}{x}=a<\/code>\uff08<code>a<\/code>\u4e3a\u975e\u96f6\u5e38\u6570\uff09\uff0c\u4e14<code class=\"katex-inline\">\\lim_{x\\to\\infty}[f(x)-ax]=b<\/code>\uff0c\u90a3\u4e48<code>y=ax+b<\/code>\u5c31\u662f<code>y=f(x)<\/code>\u7684\u659c\u6e10\u8fd1\u7ebf\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E6%9B%B2%E7%BA%BF%E5%92%8C%E5%BC%A7%E5%BE%AE%E5%88%86%E5%92%8C%E6%9B%B2%E7%8E%87\"><\/span>\u66f2\u7ebf\u548c\u5f27\u5fae\u5206\u548c\u66f2\u7387<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8bbe<code>y=f(x)<\/code>\u5728<code>(a,b)<\/code>\u5185\u6709\u8fde\u7eed\u5bfc\u6570\uff0c\u5219\u6709<strong>\u5f27\u5fae\u5206<\/strong>\uff1a<code class=\"katex-inline\">ds=\\sqrt{1+y'^2}dx<\/code><br \/>\n\u8bbe<code>y=f(x)<\/code>\u6709\u4e8c\u9636\u5bfc\u6570\uff0c\u5219\u6709<strong>\u66f2\u7387<\/strong>\uff1a<code class=\"katex-inline\">K=\\frac{|y''|}{(1+y'^2)^{\\frac{3}{2}}}<\/code><br \/>\n\u79f0<code class=\"katex-inline\">\\rho=\\frac{1}{K}<\/code>\u4e3a\u66f2\u7387\u534a\u5f84\u3002<\/p>\n<p>\u82e5\u66f2\u7ebf<code>y=f(x)<\/code>\u5728\u70b9<code>M(x,y)<\/code>\u5904\u7684\u66f2\u7387\u4e3a<code class=\"katex-inline\">K\uff08K\\ne0\uff09<\/code>\u3002\u5728\u70b9<code>M<\/code>\u5904\u66f2\u7ebf\u7684\u6cd5\u7ebf\u4e0a\uff0c\u5728\u66f2\u7ebf\u51f9\u7684\u4e00\u4fa7\u53d6\u4e00\u70b9<code>D<\/code>\uff0c\u4f7f<code class=\"katex-inline\">|DM|=\\frac{1}{K}=\\rho<\/code>\uff0c\u4ee5<code>D<\/code>\u4e3a\u5706\u5fc3\uff0c<code class=\"katex-inline\">\\rho<\/code>\u4e3a\u534a\u5f84\u7684\u5706\u79f0\u4e3a\u66f2\u7ebf\u5728\u70b9<code>M<\/code>\u5904\u7684<strong>\u66f2\u7387\u5706<\/strong>\uff0c\u5706\u5fc3<code>D<\/code>\u79f0\u4e3a\u66f2\u7ebf\u5728\u70b9<code>M<\/code>\u5904\u7684<strong>\u66f2\u7387\u4e2d\u5fc3<\/strong>\u3002<\/p>\n<hr \/>\n<p>\u63a8\u5bfc\u4e0e\u7406\u89e3<\/p>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/08\/image-1692973825282.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 310px; --smush-placeholder-aspect-ratio: 310\/289;\" \/><br \/>\n\u5728\u4e00\u6bb5\u5f27\u4e0a\u53d6\u4e00\u6bb5\u5c0f\u5f27\uff0c\u8fd9\u6bb5\u5c0f\u5f27\u8db3\u591f\u5c0f\uff0c\u5373<code>dx<\/code>\u8db3\u591f\u5c0f\uff0c\u90a3\u4e48\u8fd9\u6bb5\u5c0f\u5f27\u7684\u957f\u5ea6\u5c31\u53ef\u4ee5\u8fd1\u4f3c\u770b\u4f5c\u5c0f\u5f27\u4e24\u7aef\u70b9\u76f4\u7ebf\u8fde\u63a5\u7684\u8ddd\u79bb\uff0c\u5bf9\u5e94<code class=\"katex-inline\">ds=\\sqrt{(dx)^2+(dy)^2}=\\sqrt{(dx)^2+(y'\\cdot dx)^2}=\\sqrt{1+y'^2}dx<\/code>\uff0c\u8fd9\u5c31\u662f<strong>\u5f27\u5fae\u5206<\/strong>\u3002<\/p>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/08\/image-1692973855871.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 329px; --smush-placeholder-aspect-ratio: 329\/292;\" \/><br \/>\n\u5f27 <code>AB<\/code> \u548c\u5f27 <code>CD<\/code> \u957f\u5ea6\u76f8\u7b49\uff0c\u5f27 <code>AB<\/code> \u4ece <code>A<\/code> \u70b9\u8d77\u59cb\u5230 <code>B<\/code> \u70b9\u7ed3\u675f\uff0c\u5207\u7ebf\u65cb\u8f6c\u7684\u89d2\u5ea6\u5c31\u662f<code class=\"katex-inline\">\\alpha_1<\/code>\uff0c\u5f27 <code>CD<\/code> \u5207\u7ebf\u65cb\u8f6c\u7684\u89d2\u5ea6\u4e3a<code class=\"katex-inline\">\\alpha_2<\/code>\uff0c\u4ece\u76f8\u540c\u5f27\u957f\u5207\u7ebf\u65cb\u8f6c\u7684\u89d2\u5ea6\u5c31\u80fd\u4f53\u73b0\u51fa\u8fd9\u4e2a\u5f27\u7684\u5f2f\u66f2\u7a0b\u5ea6\u3002\u90a3\u4e48\u8fd9\u5c31\u53ef\u4ee5\u5f15\u51fa\u66f2\u7387\u7684\u5b9a\u4e49\uff0c\u4e5f\u5c31\u662f\u5355\u4f4d\u5f27\u957f\u5207\u7ebf\u65b9\u5411\u89d2\u7684\u8f6c\u52a8\u5927\u5c0f\u3002\u6709<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\ny&#039;=\\frac{dy}{dx}\uff0c\u51e0\u4f55\u610f\u4e49\u4e0a\u53cd\u6620\u66f2\u7ebf\u4e0a\u67d0\u70b9\u7684\u5207\u7ebf\u659c\u7387\uff0c\u90a3\u4e48\u53ef\u4ee5\u8868\u793a\u4e3a y&#039;=\\tan\\theta\uff0c\\theta \u4e3a\u8be5\u70b9\u7684\u5207\u7ebf\u4e0e x \u8f74\u7684\u5939\u89d2\u3002 \\\\\n\\theta=\\arctan y&#039; \\\\\nd\\theta=\\frac{y&#039;&#039;}{1+y&#039;^2}dx \\\\\n\u5219\u66f2\u7387 K=|\\frac{d\\theta}{ds}|=\\frac{|y&#039;&#039;|}{(1+y&#039;^2)^{\\frac{3}{2}}}\n\\end{array}<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/08\/image-1692977793352.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 263px; --smush-placeholder-aspect-ratio: 263\/259;\" \/><br \/>\n\u66f2\u7387\u5706\u662f\u53cd\u6620\u4e86\u67d0\u70b9\u5904\u6700\u63a5\u8fd1\u7684\u8fd1\u4f3c\u5706\uff0c\u5176\u534a\u5f84\u4e3a\u66f2\u7387\u7684\u5012\u6570\uff0c\u66f2\u7387\u534a\u5f84\u8d8a\u5c0f\u66f2\u7ebf\u8d8a\u63a5\u8fd1\u5706\uff0c\u8d8a\u5927\u8d8a\u63a5\u8fd1\u76f4\u7ebf\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E7%9B%B8%E5%85%B3%E5%8F%98%E5%8C%96%E7%8E%87\"><\/span>\u76f8\u5173\u53d8\u5316\u7387<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8bbe<code>x=x(t)<\/code>\uff0c<code>y=y(t)<\/code>\u90fd\u662f\u53ef\u5bfc\u51fd\u6570\uff0c\u800c\u53d8\u91cf<code>x<\/code>\u4e0e<code>y<\/code>\u4e4b\u95f4\u5b58\u5728\u67d0\u79cd\u5173\u7cfb\uff0c\u56e0\u6b64\u53d8\u5316\u7387<code class=\"katex-inline\">\\frac{dx}{dt}<\/code>\u4e0e<code class=\"katex-inline\">\\frac{dy}{dt}<\/code>\u4e5f\u5b58\u5728\u4e00\u5b9a\u5173\u7cfb\uff0c\u8fd9\u4e24\u4e2a\u76f8\u4e92\u4f9d\u8d56\u7684\u53d8\u5316\u7387\u79f0\u4e3a\u76f8\u5173\u53d8\u5316\u7387\u3002<\/p>\n<hr \/>\n<p>\u4f8b<br \/>\n\uff082016\uff0c\u6570\u4e8c\uff09\u5df2\u77e5\u52a8\u70b9<code>P<\/code>\u5728\u66f2\u7ebf<code class=\"katex-inline\">y=x^3<\/code>\u4e0a\u8fd0\u52a8\uff0c\u8bb0\u5750\u6807\u539f\u70b9\u4e0e<code>P<\/code>\u95f4\u7684\u8ddd\u79bb\u4e3a<code>l<\/code>\u3002\u70b9<code>P<\/code>\u6a2a\u5750\u6807\u65f6\u95f4\u7684\u53d8\u5316\u7387\u4e3a\u5e38\u6570<code class=\"katex-inline\">v_0<\/code>\uff0c\u5219\u5f53\u70b9<code>P<\/code>\u8fd0\u52a8\u5230\u70b9<code>(1,1)<\/code>\u65f6\uff0c<code>l<\/code>\u5bf9\u65f6\u95f4\u7684\u53d8\u5316\u7387\u662f\uff1f<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nP(x,x^3) \\\\\nl=\\sqrt{x^2+y^2}=\\sqrt{x^2+x^6} \\\\\n\\frac{dl}{dx}=\\frac{x+3x^5}{\\sqrt{x^2+x^6}} \\\\\n\\frac{dx}{dt}=v_0 \\\\\n\\frac{dl}{dt}|_{(x,y)=(1,1)}=\\frac{dl}{dx}\\cdot\\frac{dx}{dt}|_{(x,y)=(1,1)}=\\frac{x+3x^5}{\\sqrt{x^2+x^6}}\\cdot v_0|_{(x,y)=(1,1)}=2\\sqrt{2}v_0\n\\end{array}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>\u51fd\u6570\u7684\u5355\u8c03\u6027 \u8bbef(x)\u5728[a,b]\u5185\u8fde\u7eed\uff0c\u5728(a,b)\u5185\u53ef\u5bfc\u3002 \u2460\u82e5\u5728(a,b)\u5185\uff0cf'(x)\\gt0\uff0c\u5219f [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1],"tags":[],"class_list":["post-10934","post","type-post","status-publish","format-standard","hentry","category-all"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10934","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10934"}],"version-history":[{"count":1,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10934\/revisions"}],"predecessor-version":[{"id":21453,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10934\/revisions\/21453"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10934"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10934"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10934"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}