{"id":35366,"date":"2025-01-01T13:00:06","date_gmt":"2025-01-01T13:00:06","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35366"},"modified":"2025-12-14T22:47:23","modified_gmt":"2025-12-14T22:47:23","slug":"%e5%87%bd%e6%95%b0%e7%9a%84%e6%9c%80%e5%a4%a7%e5%80%bc%e4%b8%8e%e6%9c%80%e5%b0%8f%e5%80%bc","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e5%87%bd%e6%95%b0%e7%9a%84%e6%9c%80%e5%a4%a7%e5%80%bc%e4%b8%8e%e6%9c%80%e5%b0%8f%e5%80%bc\/","title":{"rendered":"\u51fd\u6570\u7684\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u51fd\u6570\u7684\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c<\/h1>\n<p style=\"text-align:center\"><em>\u51fd\u6570\u7684\u201c\u6700\u4f73\u201d\u70b9\u5728\u54ea\u91cc\uff1a\u4f60\u5e0c\u671b\u8fbe\u5230\u7684\u6700\u5927\u503c\uff0c\u8fd8\u662f\u9700\u8981\u907f\u514d\u7684\u6700\u5c0f\u503c\uff1f\u8fd9\u4e00\u95ee\u9898\u51fa\u73b0\u5728\u4f18\u5316\u3001\u7269\u7406\u3001\u7ecf\u6d4e\u5b66\u4e0e\u5de5\u7a0b\u5b66\u4e4b\u4e2d\uff0c\u662f\u5fae\u5206\u5b66\u7684\u4e3b\u8981\u5e94\u7528\u4e4b\u4e00\u3002\u800c\u5173\u952e\u5728\u4e8e\uff1a<a href=\"https:\/\/toposuranos.com\/material\/zh\/%e9%ad%8f%e5%b0%94%e6%96%af%e7%89%b9%e6%8b%89%e6%96%af%e6%9e%81%e5%80%bc%e5%ae%9a%e7%90%86\/\" rel=\"noopener\" target=\"_blank\">\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406<\/a>\u4fdd\u8bc1\uff0c\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u8fde\u7eed\uff0c\u5e76\u4e14\u7814\u7a76\u5bf9\u8c61\u662f\u4e00\u4e2a\u95ed\u4e14\u6709\u754c\u7684\u533a\u95f4\uff0c\u90a3\u4e48 <strong>\u7edd\u5bf9\u6781\u503c\u5fc5\u7136\u5b58\u5728<\/strong>\u3002\u7531\u6b64\uff0c\u95ee\u9898\u4fbf\u8f6c\u5411\u5b9e\u8df5\u5c42\u9762\uff1a\u5b66\u4e60\u5982\u4f55\u901a\u8fc7 <strong>\u4e34\u754c\u70b9<\/strong>\uff08<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x)=0<\/span><\/span> \u6216\u4e0d\u5b58\u5728\uff09\u6765\u8bc6\u522b <strong>\u5c40\u90e8\u6781\u503c<\/strong>\uff0c\u5e76\u5229\u7528\u7f57\u5c14\u5b9a\u7406\u4e0e\u5e73\u5747\u503c\u5b9a\u7406\u7b49\u5de5\u5177\uff0c\u5c06\u201c\u76f2\u76ee\u201d\u7684\u641c\u7d22\u8f6c\u5316\u4e3a\u6e05\u6670\u3001\u53ef\u9a8c\u8bc1\u4e14\u9ad8\u6548\u7684\u65b9\u6cd5\u3002<\/em><\/p>\n<p style=\"text-align:center\">\n<strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong>\n<\/p>\n<ol>\n<li><b>\u6267\u884c<\/b> \u4e00\u4e2a\u5b8c\u6574\u7684\u8fc7\u7a0b\uff0c\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u6c42\u7edd\u5bf9\u6781\u503c\uff1a\u5728\u533a\u95f4\u5185\u90e8\u7684\u4e34\u754c\u70b9\u4ee5\u53ca\u533a\u95f4\u7aef\u70b9\u5904\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u7684\u503c\uff0c\u5e76\u901a\u8fc7\u6bd4\u8f83\u8fd9\u4e9b\u503c\u6765\u5224\u5b9a\u7edd\u5bf9\u6700\u5927\u503c\u4e0e\u7edd\u5bf9\u6700\u5c0f\u503c\u3002\n  <\/li>\n<li><strong>\u5bf9\u6bd4<\/strong> \u5fc5\u8981\u6761\u4ef6\u4e0e\u5145\u5206\u6761\u4ef6\u7684\u4f5c\u7528\uff1a\u8ba4\u8bc6\u5230\u201c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x_0)=0<\/span><\/span>\u201d\u5e76\u4e0d\u80fd\u4fdd\u8bc1\u5c40\u90e8\u6781\u503c\u7684\u5b58\u5728\uff0c\u5e76\u5224\u65ad\u5728\u4e0d\u540c\u60c5\u5f62\u4e0b\u9700\u8981\u54ea\u4e9b\u9644\u52a0\u8bc1\u636e\uff08\u6570\u503c\u6bd4\u8f83\u3001\u7b26\u53f7\u5206\u6790\u3001\u5c40\u90e8\u884c\u4e3a\uff09\u3002\n  <\/li>\n<li><strong>\u786e\u5b9a<\/strong> \u9488\u5bf9\u4e0d\u540c\u7c7b\u578b\u95ee\u9898\u7684\u6700\u6709\u6548\u7b56\u7565\uff1a\u7d27\u533a\u95f4\u4e0a\u7684\u7edd\u5bf9\u6781\u503c\uff08\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406 + \u6709\u9650\u6b21\u8ba1\u7b97\uff09\u4e0e\u5185\u90e8\u70b9\u7684\u5c40\u90e8\u6781\u503c\uff08\u4e34\u754c\u70b9 + \u5c40\u90e8\u5206\u6790\uff09\uff0c\u5e76\u5bf9\u7b56\u7565\u9009\u62e9\u7ed9\u51fa\u5408\u7406\u8bba\u8bc1\u3002\n  <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<strong><u>\u5185\u5bb9\u7d22\u5f15<\/u>\uff1a<\/strong><br \/>\n<a href=\"#1\"><strong>\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\uff0c\u7edd\u5bf9\u6781\u503c\u4e0e\u5c40\u90e8\u6781\u503c<\/strong><\/a><br \/>\n<a href=\"#2\">\u4e00\u9636\u5bfc\u6570\u5224\u522b\u6cd5<\/a><br \/>\n<a href=\"#3\"><strong>\u7f57\u5c14\u5b9a\u7406<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>\u5fae\u5206\u4e2d\u503c\u5b9a\u7406<\/strong><\/a><br \/>\n<a href=\"#5\">\u51fd\u6570\u7684\u589e\u533a\u95f4\u4e0e\u51cf\u533a\u95f4<\/a>\n<\/p>\n<div style=\"text-align: center;\">\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/OscTlX3raaE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<\/div>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/toposuranos.com\/material\/zh\/%e9%ad%8f%e5%b0%94%e6%96%af%e7%89%b9%e6%8b%89%e6%96%af%e6%9e%81%e5%80%bc%e5%ae%9a%e7%90%86\/\" rel=\"noopener\" target=\"_blank\">\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406<\/a>\u6307\u51fa\uff0c\u5982\u679c\u4e00\u4e2a\u5b9e\u51fd\u6570\u5b9a\u4e49\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span> \u7684\u67d0\u4e2a\u95ed\u4e14\u6709\u754c\u5b50\u96c6\u4e0a\u5e76\u4e14\u662f\u8fde\u7eed\u7684\uff0c\u90a3\u4e48\u5b83\u5fc5\u7136\u53d6\u5f97\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c\uff08\u7edd\u5bf9\u6781\u503c\uff09\u3002\u5bfb\u627e\u51fd\u6570\u7684\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c\u88ab\u79f0\u4e3a <strong>\u4f18\u5316\u95ee\u9898<\/strong>\uff0c\u800c\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406\u4fdd\u8bc1\u4e86\u5728\u7edd\u5bf9\u6781\u503c\u610f\u4e49\u4e0b\u89e3\u7684\u5b58\u5728\u6027\uff0c\u53ea\u8981\u51fd\u6570\u8fde\u7eed\u4e14\u5b9a\u4e49\u57df\u662f\u7d27\u96c6\u3002\u5728\u5b58\u5728\u6027\u5f97\u5230\u4fdd\u8bc1\u4e4b\u540e\uff0c\u5269\u4e0b\u7684\u4efb\u52a1\u5c31\u662f\u53d1\u5c55\u80fd\u591f\u627e\u5230\u8fd9\u4e9b\u89e3\u7684\u5177\u4f53\u7b56\u7565\u3002\n<\/p>\n<p><a name=\"1\"><\/a><\/br><\/p>\n<h2>\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\uff0c\u7edd\u5bf9\u6781\u503c\u4e0e\u5c40\u90e8\u6781\u503c<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=156s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u5728\u5f00\u59cb\u56de\u987e<\/span><\/a> \u5bfb\u627e\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\u7684\u7b56\u7565\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u6e05\u6670\u5730\u754c\u5b9a\u6211\u4eec\u7a76\u7adf\u8981\u5bfb\u627e\u7684\u5bf9\u8c61\u3002<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u5b9a\u4e49\uff1a<\/strong><\/span><br \/>\n\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u4e00\u4e2a\u5b9a\u4e49\u57df\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span> \u7684\u51fd\u6570\u3002\u82e5\u5b58\u5728\u4e00\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span>\uff0c\u4f7f\u5f97\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( \\forall x \\in D \\right)\\bigl(f(x) \\leq f(x_0)\\bigr)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u5219\u79f0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728\u8be5\u70b9\u53d6\u5f97\u4e00\u4e2a <strong>\u7edd\u5bf9\u6700\u5927\u503c<\/strong>\uff1b\u7c7b\u4f3c\u5730\uff0c\u82e5\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u6ee1\u8db3\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( \\forall x \\in D \\right)\\bigl( f(x_0) \\leq f(x)\\bigr)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u5219\u79f0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728\u8be5\u70b9\u53d6\u5f97\u4e00\u4e2a <strong>\u7edd\u5bf9\u6700\u5c0f\u503c<\/strong>\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  \u4ee5\u7c7b\u4f3c\u7684\u65b9\u5f0f\uff0c\u53ef\u4ee5\u5b9a\u4e49\u5c40\u90e8\u6781\u503c\uff08\u76f8\u5bf9\u4e8e\u5b9a\u4e49\u57df\u800c\u8a00\uff09\u3002\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u5b9a\u4e49\uff1a<\/strong><\/span><br \/>\n          \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u4e00\u4e2a\u5b9a\u4e49\u57df\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span> \u7684\u51fd\u6570\uff0c\u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span>\u3002\u82e5\u5b58\u5728\u67d0\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h&gt;0<\/span><\/span>\uff0c\u4f7f\u5f97\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\exists h&gt;0)\\left( \\forall x\\in [x_0-h, x_0+h] \\cap D \\right)\\bigl(f(x) \\leq f(x_0)\\bigr)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u5219\u79f0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u5904\u53d6\u5f97\u4e00\u4e2a <strong>\u5c40\u90e8\u6700\u5927\u503c<\/strong>\uff1b\u7c7b\u4f3c\u5730\uff0c\u82e5\u6ee1\u8db3\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\exists h&gt;0)\\left( \\forall x\\in [x_0-h, x_0+h] \\cap D \\right)\\bigl( f(x_0) \\leq f(x)\\bigr)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u5219\u79f0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u5904\u53d6\u5f97\u4e00\u4e2a <strong>\u5c40\u90e8\u6700\u5c0f\u503c<\/strong>\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  \u7531\u6b64\u6211\u4eec\u53ef\u4ee5\u8868\u8ff0\u5982\u4e0b\u7ed3\u679c\uff1a\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>\u5b9a\u7406\uff1a<\/strong><\/span><br \/>\n          <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=833s\" target=\"_blank\" rel=\"noopener\"><br \/>\n            <strong><span style=\"color: #ff0000;\">\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u662f\u4e00\u4e2a\u70b9<\/span><\/strong><\/a>\uff0c\u4f4d\u4e8e\u67d0\u4e2a\u7d27\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/span> \u7684\u5185\u90e8\u3002\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u5904\u53d6\u5f97\u4e00\u4e2a <strong>\u5c40\u90e8<\/strong> \u6700\u5927\u503c\u6216\u6700\u5c0f\u503c\uff0c\u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> \u5b58\u5728\uff0c\u5219\u5fc5\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)=0<\/span><\/span>\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/span><br \/>\n          \u5047\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u5904\u53d6\u5f97\u4e00\u4e2a\u5c40\u90e8\u6700\u5927\u503c\u3002\u5219\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h_0 \\gt 0<\/span><\/span>\uff0c\u4f7f\u5f97\u5bf9\u6240\u6709\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|\\lt h_0<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0+h\\in I<\/span><\/span> \u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h<\/span><\/span>\uff0c\u90fd\u6709\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0 + h)\\leq f(x_0)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u8fd9\u7b49\u4ef7\u4e8e\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0 + h) - f(x_0)\\leq 0<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u73b0\u5728\u8003\u8651\u4e24\u79cd\u60c5\u51b5\uff1a\n        <\/p>\n<ul>\n<li>\n<p style=\"text-align: justify;\">\n              \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h&gt;0<\/span><\/span>\uff0c\u5219\uff1a\n            <\/p>\n<p style=\"text-align: center;\">\n              <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{f(x_0 + h) - f(x_0)}{h}\\leq 0<\/span><\/span>\n            <\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\">\n              \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\lt 0<\/span><\/span>\uff0c\u5219\uff1a\n            <\/p>\n<p style=\"text-align: center;\">\n              <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{f(x_0 + h) - f(x_0)}{h}\\geq 0<\/span><\/span>\n            <\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\n          \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> \u5b58\u5728\uff0c\u5219\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\to 0<\/span><\/span> \u65f6\uff0c\u589e\u91cf\u5546\u7684\u6781\u9650\u5b58\u5728\uff0c\u5e76\u4e14\u5fc5\u987b\u540c\u65f6\u6ee1\u8db3\u4e0a\u8ff0\u4e24\u79cd\u4e0d\u7b49\u5f0f\uff0c\u8fd9\u5c31\u8feb\u4f7f\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f^\\prime(x_0)=\\lim_{h\\to 0}\\frac{f(x_0 + h) - f(x_0)}{h}= 0<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u8fd9\u6b63\u662f\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u8be5\u8bc1\u660e\u5bf9\u5c40\u90e8\u6700\u5c0f\u503c\u540c\u6837\u6210\u7acb\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8d77\u59cb\u6761\u4ef6\u4e3a\uff1a\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|<\/span><\/span> \u8db3\u591f\u5c0f\u65f6\uff0c\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0+h)\\ge f(x_0)<\/span><\/span>\u3002\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h3>\u4e00\u9636\u5bfc\u6570\u5224\u522b\u6cd5<\/h3>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=1257s\" target=\"_blank\" rel=\"noopener\"><br \/>\n    <strong><span style=\"color: #ff0000;\">\u6211\u4eec\u521a\u521a\u56de\u987e\u7684\u7ed3\u679c<\/span><\/strong><\/a> \u53ef\u4ee5\u6982\u62ec\u4e3a\u5982\u4e0b\u8574\u542b\u5173\u7cfb\uff1a<\/p>\n<p style=\"text-align: center;\">\n  <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  \\left\\{\\begin{matrix}f \\text{ \u5728 }x_0\\text{ \u5904\u53d6\u5f97}\\\\ \\text{\u4e00\u4e2a\u5c40\u90e8\u6781\u503c} \\end{matrix}\\right\\}\n\n  \\Longrightarrow\n\n  \\left\\{\\begin{matrix} \\displaystyle f^\\prime(x_0) = 0 \\\\ \\\\ \\vee \\\\ \\\\ \\text{\u5bfc\u6570\u5728 }x_0\\text{ \u5904\u4e0d\u5b58\u5728} \\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  \u5c3d\u7ba1\u8be5\u8574\u542b\u5173\u7cfb\u7684\u9006\u547d\u9898\u5728\u4e00\u822c\u60c5\u51b5\u4e0b\u5e76\u4e0d\u6210\u7acb\uff0c\u4f46\u5b83\u5728\u7f29\u5c0f\u5c40\u90e8\u6781\u503c\u641c\u7d22\u8303\u56f4\u65f6\u975e\u5e38\u6709\u7528\u3002\u57fa\u4e8e\u8fd9\u4e00\u70b9\uff0c\u5f15\u5165\u4e86\u4e00\u9636\u5bfc\u6570\u7684\u4e34\u754c\u70b9\u6982\u5ff5\u3002\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u5b9a\u4e49\uff1a<\/strong><\/span><br \/>\n          \u79f0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u4e3a <strong>\u4e00\u9636\u5bfc\u6570\u7684\u4e34\u754c\u70b9<\/strong>\uff0c\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)=0<\/span><\/span>\uff0c\u6216\u8005 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> \u5728\u8be5\u70b9\u4e0d\u5b58\u5728\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  \u4e00\u9636\u5bfc\u6570\u7684\u4e34\u754c\u70b9\u4e4b\u6240\u4ee5\u91cd\u8981\uff0c\u662f\u56e0\u4e3a\u51fd\u6570\u53d6\u5f97\u6781\u503c\uff08\u65e0\u8bba\u662f\u5c40\u90e8\u6781\u503c\u8fd8\u662f\u7edd\u5bf9\u6781\u503c\uff09\u7684\u4efb\u4f55\u70b9\uff0c\u90fd\u5fc5\u987b\u5c5e\u4e8e\u4e34\u754c\u70b9\u7684\u96c6\u5408\uff1a\n<\/p>\n<p style=\"text-align: center;\">\n  <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left\\{\\begin{matrix}\\text{\u53d6\u5f97\u7edd\u5bf9\u6781\u503c\u7684}\\\\ \\text{\u70b9}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{\u53d6\u5f97\u5c40\u90e8\u6781\u503c\u7684}\\\\ \\text{\u70b9}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{\u4e00\u9636\u5bfc\u6570\u7684}\\\\ \\text{\u4e34\u754c\u70b9}\\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  \u8fd9\u5c31\u662f\u6211\u4eec\u6240\u79f0\u7684 <strong>\u4e00\u9636\u5bfc\u6570\u5224\u522b\u6cd5<\/strong>\uff0c\u5176\u88ab\u7406\u89e3\u4e3a\u5185\u70b9\u5904\u5b58\u5728\u5c40\u90e8\u6781\u503c\u7684\u4e00\u4e2a\u5fc5\u8981\u6761\u4ef6\u3002\n<\/p>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>\u7f57\u5c14\u5b9a\u7406<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=1454s\" target=\"_blank\" rel=\"noopener\"><br \/>\n    <strong><span style=\"color: #ff0000;\">\u6211\u4eec\u5df2\u7ecf\u770b\u5230\uff0c\u4e00\u9636\u5bfc\u6570\u4e34\u754c\u70b9\u7684\u786e\u5b9a<\/span><\/strong><\/a> \u5728\u5bfb\u627e\u5c40\u90e8\u6781\u503c\u7684\u8fc7\u7a0b\u4e2d\u8d77\u7740\u5173\u952e\u4f5c\u7528\u3002\u56e0\u6b64\uff0c\u81ea\u7136\u4f1a\u8fdb\u4e00\u6b65\u63a2\u7a76\u5728\u4f55\u79cd\u6761\u4ef6\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u8fd9\u4e9b\u4e34\u754c\u70b9\u7684\u5b58\u5728\u3002\u5728\u8fd9\u4e00\u65b9\u5411\u4e0a\u7684\u91cd\u8981\u8fdb\u5c55\u6765\u81ea\u7f57\u5c14\u5b9a\u7406\u3002<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>\u5b9a\u7406\uff1a<\/strong><\/span><br \/>\n          \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u5b9a\u4e49\u5e76\u8fde\u7eed\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u3001\u4e14\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u53ef\u5bfc\u7684\u51fd\u6570\u3002\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span>\uff0c\u5219\u5b58\u5728\u67d0\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/span><br \/>\n          \u6211\u4eec\u5206\u6790\u4ee5\u4e0b\u4e24\u79cd\u53ef\u80fd\u60c5\u51b5\uff1a\n        <\/p>\n<ol style=\"text-align: justify;\">\n<li>\n            \u82e5\u5bf9\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> \u90fd\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=f(a)=f(b)<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u4e3a\u5e38\u51fd\u6570\uff0c\u56e0\u6b64\u5bf9\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> \u90fd\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x)=0<\/span><\/span>\u3002\u7279\u522b\u5730\uff0c\u5b58\u5728\u67d0\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>\u3002\n          <\/li>\n<li>\n            \u82e5\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)\\neq f(a)=f(b)<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u975e\u5e38\u51fd\u6570\u3002\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u8fde\u7eed\uff0c\u6839\u636e\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406\uff0c\u5b83\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u53d6\u5f97\u7edd\u5bf9\u6700\u5927\u503c\u548c\u7edd\u5bf9\u6700\u5c0f\u503c\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u4e0d\u662f\u5e38\u51fd\u6570\uff0c\u8fd9\u4e24\u4e2a\u6781\u503c\u4e2d\u81f3\u5c11\u6709\u4e00\u4e2a\u53d1\u751f\u5728\u533a\u95f4\u5185\u90e8 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> \u662f\u4e00\u4e2a\u4f7f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u53d6\u5f97\u5c40\u90e8\u6781\u503c\u7684\u5185\u70b9\u3002\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u53ef\u5bfc\uff0c\u7279\u522b\u5730 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)<\/span><\/span> \u5b58\u5728\uff0c\u7531\u524d\u8ff0\u5b9a\u7406\u53ef\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>\u3002\n          <\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/p>\n<h2>\u5fae\u5206\u4e2d\u503c\u5b9a\u7406<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=1878s\" target=\"_blank\" rel=\"noopener\"><br \/>\n    <strong><span style=\"color: #ff0000;\">\u53e6\u4e00\u4e2a\u4f5c\u4e3a\u6211\u4eec\u521a\u521a\u6240\u56de\u987e\u7ed3\u679c\u7684\u76f4\u63a5\u63a8\u8bba<\/span><\/strong><\/a>\uff0c\u5e76\u4e14\u4e3a\u51fd\u6570\u7814\u7a76\u63d0\u4f9b\u6709\u7528\u4fe1\u606f\u7684\u7ed3\u679c\uff0c\u662f\u5fae\u5206\u5b66\u4e2d\u7684\u4e2d\u503c\u5b9a\u7406\u3002\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>\u5b9a\u7406\uff1a<\/strong><\/span><br \/>\n          \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u4e00\u4e2a\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u5b9a\u4e49\u4e14\u8fde\u7eed\uff0c\u5e76\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u53ef\u5bfc\u7684\u51fd\u6570\u3002\u5219\u5b58\u5728\u67d0\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span>\uff0c\u4f7f\u5f97\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) =\\displaystyle \\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/span><br \/>\n          \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u4e3a\u5982\u4e0b\u5b9a\u4e49\u7684\u51fd\u6570\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x) = f(x) - \\displaystyle \\frac{f(b) - f(a)}{b-a}(x-a)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5177\u6709\u76f8\u5e94\u6027\u8d28\uff0c\u8be5\u51fd\u6570\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u4e0a\u8fde\u7eed\uff0c\u5e76\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u53ef\u5bfc\u3002\u6b64\u5916\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a)=F(b)<\/span><\/span>\uff0c\u56e0\u6b64\u53ef\u4ee5\u5e94\u7528\u7f57\u5c14\u5b9a\u7406\uff0c\u4ece\u800c\u5f97\u51fa\u5b58\u5728\u67d0\u4e2a\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F^\\prime(c)=0<\/span><\/span>\u3002\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u63a5\u4e0b\u6765\uff0c\u5bf9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u6c42\u5bfc\uff0c\u5f97\u5230\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F^\\prime(x) = f^\\prime(x) - \\displaystyle\\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u5904\u53d6\u503c\uff0c\u5e76\u5229\u7528 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F^\\prime(c)=0<\/span><\/span>\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=F^\\prime(c) = f^\\prime(c) - \\displaystyle\\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u56e0\u6b64\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) = \\displaystyle\\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u8fd9\u6b63\u662f\u6240\u8981\u8bc1\u660e\u7684\u7ed3\u8bba\u3002\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><\/br><\/p>\n<h3>\u589e\u957f\u533a\u95f4\u4e0e\u9012\u51cf\u533a\u95f4<\/h3>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>\u5b9a\u7406\uff1a<\/strong><\/span>\n        <\/p>\n<ul style=\"text-align: justify;\">\n<li>\n            <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=2402s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u4e00\u4e2a\u51fd\u6570<\/span><\/strong><\/a>\uff0c\u5e76\u4e14\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in ]a,b[)\\left(0\\lt f^\\prime(x)\\right)<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u4e25\u683c\u9012\u589e\u3002\n          <\/li>\n<li>\n            \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u4e00\u4e2a\u51fd\u6570\uff0c\u5e76\u4e14\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in ]a,b[)\\left(f^\\prime(x)\\lt 0\\right)<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u4e25\u683c\u9012\u51cf\u3002\n          <\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/span><br \/>\n          \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1,x_2\\in ]a,b[<\/span><\/span>\uff0c\u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1 \\lt x_2<\/span><\/span>\u3002\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> \u4e0a\u53ef\u5bfc\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[x_1,x_2]\\subset ]a,b[<\/span><\/span> \u4e0a\u5bf9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u5e94\u7528\u5fae\u5206\u4e2d\u503c\u5b9a\u7406\u3002\u56e0\u6b64\uff0c\u5b58\u5728\u67d0\u4e2a\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]x_1,x_2[<\/span><\/span>\uff0c\u4f7f\u5f97\uff1a\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) = \\displaystyle\\frac{f(x_2) - f(x_1)}{x_2 - x_1}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          \u7531\u6b64\u53ef\u5f97\uff1a\n        <\/p>\n<ul style=\"text-align: justify;\">\n<li>\n            \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\gt 0<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_2) - f(x_1) = f^\\prime(c)(x_2 - x_1) \\gt 0<\/span><\/span>\u3002<br \/>\n\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u9012\u589e\u7684\u3002\n          <\/li>\n<li>\n            \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\lt 0<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_2) - f(x_1) = f^\\prime(c)(x_2 - x_1) \\lt 0<\/span><\/span>\u3002<br \/>\n\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u662f\u9012\u51cf\u7684\u3002\n          <\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n\u7814\u7a76\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\u4e0d\u4ec5\u4ec5\u662f\u201c\u505a\u5bfc\u6570\u8fd0\u7b97\u201d\uff0c\u800c\u662f\u5b66\u4e60\u5982\u4f55\u5c06\u4e00\u79cd\u6a21\u7cca\u7684\u641c\u7d22\u8f6c\u5316\u4e3a\u4e00\u4e2a\u5177\u6709\u7406\u8bba\u4fdd\u8bc1\u548c\u660e\u786e\u5224\u636e\u7684\u8fc7\u7a0b\u3002\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406\u544a\u8bc9\u4f60\uff0c\u5728\u7d27\u533a\u95f4\u4e0a\u4f55\u65f6\u53ef\u4ee5\u786e\u4fe1\u6700\u4f18\u89e3\u7684\u5b58\u5728\uff1b\u800c\u4e00\u9636\u5bfc\u6570\u5224\u522b\u6cd5\u3001\u7f57\u5c14\u5b9a\u7406\u4ee5\u53ca\u5fae\u5206\u4e2d\u503c\u5b9a\u7406\u5219\u4e3a\u4f60\u63d0\u4f9b\u4e86\u4e00\u5f20\u8def\u7ebf\u56fe\uff0c\u7528\u4ee5\u5bfb\u627e\u5019\u9009\u70b9\u5e76\u8bba\u8bc1\u7ed3\u8bba\uff1a\u51fd\u6570\u53ef\u80fd\u5728\u4f55\u5904\u53d6\u5f97\u6781\u503c\uff0c\u4f55\u65f6\u8be5\u6761\u4ef6\u4ec5\u662f\u5fc5\u8981\u7684\uff0c\u4ee5\u53ca <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;<\/span><\/span> \u7684\u7b26\u53f7\u5982\u4f55\u63ed\u793a\u51fd\u6570\u7684\u589e\u957f\u4e0e\u9012\u51cf\u3002\u4e00\u65e6\u4f60\u638c\u63e1\u4e86\u8fd9\u4e00\u6574\u5957\u601d\u60f3\u94fe\u6761\uff0c\u4fbf\u80fd\u4ece\u51ed\u76f4\u89c9\u89c2\u5bdf\u56fe\u50cf\uff0c\u8fc7\u6e21\u5230\u57fa\u4e8e\u53ef\u9a8c\u8bc1\u8bba\u8bc1\u7684\u4f18\u5316\u6c42\u89e3\uff0c\u8fd9\u6b63\u662f\u201c\u6211\u8ba4\u4e3a\u6700\u4f73\u70b9\u5728\u8fd9\u91cc\u201d\u4e0e\u201c\u6211\u77e5\u9053\u5b83\u5fc5\u987b\u5728\u8fd9\u91cc\u201d\u4e4b\u95f4\u7684\u672c\u8d28\u5dee\u522b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u51fd\u6570\u7684\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c \u51fd\u6570\u7684\u201c\u6700\u4f73\u201d\u70b9\u5728\u54ea\u91cc\uff1a\u4f60\u5e0c\u671b\u8fbe\u5230\u7684\u6700\u5927\u503c\uff0c\u8fd8\u662f\u9700\u8981\u907f\u514d\u7684\u6700\u5c0f\u503c\uff1f\u8fd9\u4e00\u95ee\u9898\u51fa\u73b0\u5728\u4f18\u5316\u3001\u7269\u7406\u3001\u7ecf\u6d4e\u5b66\u4e0e\u5de5\u7a0b\u5b66\u4e4b\u4e2d\uff0c\u662f\u5fae\u5206\u5b66\u7684\u4e3b\u8981\u5e94\u7528\u4e4b\u4e00\u3002\u800c\u5173\u952e\u5728\u4e8e\uff1a\u9b4f\u5c14\u65af\u7279\u62c9\u65af\u5b9a\u7406\u4fdd\u8bc1\uff0c\u5982\u679c \u8fde\u7eed\uff0c\u5e76\u4e14\u7814\u7a76\u5bf9\u8c61\u662f\u4e00\u4e2a\u95ed\u4e14\u6709\u754c\u7684\u533a\u95f4\uff0c\u90a3\u4e48 \u7edd\u5bf9\u6781\u503c\u5fc5\u7136\u5b58\u5728\u3002\u7531\u6b64\uff0c\u95ee\u9898\u4fbf\u8f6c\u5411\u5b9e\u8df5\u5c42\u9762\uff1a\u5b66\u4e60\u5982\u4f55\u901a\u8fc7 \u4e34\u754c\u70b9\uff08 \u6216\u4e0d\u5b58\u5728\uff09\u6765\u8bc6\u522b \u5c40\u90e8\u6781\u503c\uff0c\u5e76\u5229\u7528\u7f57\u5c14\u5b9a\u7406\u4e0e\u5e73\u5747\u503c\u5b9a\u7406\u7b49\u5de5\u5177\uff0c\u5c06\u201c\u76f2\u76ee\u201d\u7684\u641c\u7d22\u8f6c\u5316\u4e3a\u6e05\u6670\u3001\u53ef\u9a8c\u8bc1\u4e14\u9ad8\u6548\u7684\u65b9\u6cd5\u3002 \u5b66\u4e60\u76ee\u6807\uff1a \u6267\u884c \u4e00\u4e2a\u5b8c\u6574\u7684\u8fc7\u7a0b\uff0c\u5728 \u4e0a\u6c42\u7edd\u5bf9\u6781\u503c\uff1a\u5728\u533a\u95f4\u5185\u90e8\u7684\u4e34\u754c\u70b9\u4ee5\u53ca\u533a\u95f4\u7aef\u70b9\u5904\u8ba1\u7b97 \u7684\u503c\uff0c\u5e76\u901a\u8fc7\u6bd4\u8f83\u8fd9\u4e9b\u503c\u6765\u5224\u5b9a\u7edd\u5bf9\u6700\u5927\u503c\u4e0e\u7edd\u5bf9\u6700\u5c0f\u503c\u3002 \u5bf9\u6bd4 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