{"id":2132,"date":"2023-04-02T10:02:00","date_gmt":"2023-04-02T02:02:00","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2132"},"modified":"2023-04-05T20:01:10","modified_gmt":"2023-04-05T12:01:10","slug":"fundamentals-of-advanced-mathematics-application-of-derivatives-monotonicity-concave-convex-extreme-value-and-maximum-value","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-advanced-mathematics-application-of-derivatives-monotonicity-concave-convex-extreme-value-and-maximum-value\/","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u57fa\u7840\uff1a\u5bfc\u6570\u7684\u5e94\u75281\uff1a\u5355\u8c03\u6027\u3001\u51f9\u51f8\u6027\u3001\u6781\u503c\u4e0e\u6700\u503c"},"content":{"rendered":"

\u51fd\u6570\u5355\u8c03\u6027<\/h2>\n

\u901a\u8fc7\u51fd\u6570\u7684\u5bfc\u6570\u7684\u503c\uff0c\u53ef\u4ee5\u5224\u65ad\u51fa\u51fd\u6570\u7684\u5355\u8c03\u6027\u3001\u9a7b\u70b9\u4ee5\u53ca\u6781\u503c\u70b9:<\/p>\n

\u82e5\u5bfc\u6570\u5927\u4e8e0<\/strong>\uff0c\u5219\u5355\u8c03\u9012\u589e<\/strong>\uff1b\u82e5\u5bfc\u6570\u5c0f\u4e8e0<\/strong>\uff0c\u5219\u5355\u8c03\u9012\u51cf\uff1b\u5bfc\u6570\u7b49\u4e8e0<\/strong>\u7684\u70b9\u4e3a\u51fd\u6570\u9a7b\u70b9<\/strong>\u3002<\/p>\n

<\/p>\n

\u5982\u679c\u51fd\u6570\u7684\u5bfc\u51fd\u6570\u5728\u67d0\u4e00\u4e2a\u533a\u95f4\u5185\u6052\u5927\u4e8e0<\/strong>\uff08\u6216\u6052\u5c0f\u4e8e0<\/strong>\uff09\uff0c\u90a3\u4e48\u51fd\u6570\u5728\u8fd9\u4e00\u4e2a\u533a\u95f4\u5355\u8c03\u9012\u589e\uff08\u6216\u5355\u8c03\u9012\u51cf\uff09\uff0c\u8fd9\u79cd\u533a\u95f4\u5c31\u53eb\u505a\u5355\u8c03\u533a\u95f4<\/strong>\u3002<\/p>\n

\u51fd\u6570\u7684\u9a7b\u70b9\u548c\u4e0d\u53ef\u5bfc\u70b9\u5904\u6709\u53ef\u80fd\u53d6\u5f97\u6781\u5927\u503c\u6216\u8005\u6781\u5c0f\u503c\uff08\u6781\u503c\u53ef\u7591\u70b9\uff09\uff1b\u5bf9\u4e8e\u6781\u503c\u70b9\u7684\u5224\u65ad\u9700\u8981\u5224\u65ad\u9a7b\u70b9\u9644\u8fd1\u7684\u5bfc\u51fd\u6570\u503c\u7684\u7b26\u53f7\uff0c\u5982\u679c\u5b58\u5728\u4f7f\u5f97\u4e4b\u524d\u533a\u95f4\u4e0a\u5bfc\u51fd\u6570\u503c\u90fd\u5927\u4e8e\u96f6\uff0c\u800c\u4e4b\u540e\u7684\u533a\u95f4\u4e0a\u90fd\u5c0f\u4e8e\u96f6\uff0c\u90a3\u4e48\u8fd9\u4e2a\u70b9\u5c31\u662f\u4e00\u4e2a\u6781\u5927\u503c\u70b9\uff0c\u53cd\u4e4b\u5219\u662f\u4e00\u4e2a\u6781\u5c0f\u503c\u70b9\u3002<\/p>\n

\u51fd\u6570\u51f9\u51f8\u6027<\/h2>\n

\u5b9a\u4e49\uff1a\u8bbe\u51fd\u6570$f(x)$\u5728\u533a\u95f4$I$\u4e0a\u8fde\u7eed\uff0c$\\forall x_1,x_2 \\in I$<\/p>\n

\u82e5\u6052\u6709<\/p>\n

$$
\nf(\\frac{x_1+x_2}{2}) < \\frac{f(x_1)+f(x_2)}{2}
\n$$<\/p>\n

\u5219\u79f0$f(x)$\u7684\u56fe\u5f62\u662f\u51f9\u7684\u3002<\/p>\n

\u82e5\u6052\u6709<\/p>\n

$$
\nf(\\frac{x_1+x_2}{2}) > \\frac{f(x_1)+f(x_2)}{2}
\n$$<\/p>\n

\u5219\u79f0$f(x)$\u7684\u56fe\u5f62\u662f\u51f8\u7684\u3002<\/p>\n

\u8fde\u7eed\u66f2\u7ebf\u4e0a\u51f9\u5f27\u4e0e\u51f8\u5f27\u7684\u5206\u754c\u70b9\u79f0\u4e3a\u66f2\u7ebf\u7684\u62d0\u70b9\u3002<\/p>\n

\n

\u62d0\u70b9\u662f\u5426\u5b58\u5728\uff1a\u4e8c\u9636\u5bfc\u6570\u4e3a0 \u6216 \u4e8c\u9636\u5bfc\u6570\u4e0d\u5b58\u5728\uff0c\u5177\u4f53\u8fd8\u8981\u770b\u5de6\u53f3\u4e24\u4fa7\u7684\u51f9\u51f8\u6027\u662f\u5426\u53d1\u751f\u6539\u53d8<\/p>\n<\/blockquote>\n

\u5b9a\u7406\uff08\u51f9\u51f8\u5224\u5b9a\u6cd5\uff09\uff1a\u8bbe\u51fd\u6570$f(x)$\u5728\u533a\u95f4$I$\u4e0a\u6709\u4e8c\u9636\u5bfc\u6570<\/p>\n

(1) \u5728$I$\u5185$f”(x)>0$\uff0c\u5219$f(x)$\u5728$I$\u5185\u56fe\u5f62\u662f\u51f9\u7684\uff1b
\n(1) \u5728$I$\u5185$f”(x)<0$\uff0c\u5219$f(x)$\u5728$I$\u5185\u56fe\u5f62\u662f\u51f8\u7684\uff1b<\/p>\n

\u51fd\u6570\u7684\u6781\u503c\u4e0e\u6700\u503c<\/h2>\n

\u5b9a\u4e49\uff1a\u8bbe\u51fd\u6570$f(x)$\u5728$x_0$\u7684\u67d0\u4e2a\u90bb\u57df$U(x_0,\\delta)$\u6709\u5b9a\u4e49\uff0c\u4e14\u5f53$x \\in U(x_0,\\delta)$\u65f6\uff0c\u6052\u6709$f(x)<f(x_0)$\uff0c\u5219\u79f0$f(x_0)$\u4e3a$f(x)$\u7684\u4e00\u4e2a\u6781\u5927\u503c\uff1b\u5982\u679c\u5f53$x \\in U(x_0,\\delta)$\u65f6\uff0c\u6052\u6709$f(x)>f(x_0)$\uff0c\u5219\u79f0$f(x_0)$\u4e3a$f(x)$\u7684\u4e00\u4e2a\u6781\u5c0f\u503c\u3002\u51fd\u6570\u7684\u6781\u5927\u503c\u4e0e\u6781\u5c0f\u503c\u7edf\u79f0\u4e3a\u6781\u503c\uff0c\u4f7f\u51fd\u6570\u53d6\u5f97\u6781\u503c\u7684\u70b9\u79f0\u4e3a\u6781\u503c\u70b9\u3002<\/p>\n

\u82e5$f(x)$\u5728\u6781\u503c\u70b9$x_0$\u5904\u53ef\u5bfc\uff0c\u5219$f'(x_0)=0$\u3002\u5bfc\u6570\u7b49\u4e8e$0$\u7684\u70b9\u79f0\u4e3a\u9a7b\u70b9\uff0c\u5bf9\u53ef\u5bfc\u51fd\u6570\u6765\u8bb2\uff0c\u6781\u503c\u70b9\u5fc5\u4e3a\u9a7b\u70b9\u3002<\/p>\n

\u6781\u503c\u5b58\u5728\u7684\u7b2c\u4e00\u5145\u5206\u6761\u4ef6<\/h2>\n

\u8bbe\u51fd\u6570$f(x)$\u5728$x_0$\u8fde\u7eed\uff0c\u4e14\u5728$x_0$\u7684\u67d0\u53bb\u5fc3\u90bb\u57df$U(x_0,\\delta)$\u5185\u53ef\u5bfc<\/p>\n

\uff081\uff09\u82e5\u5f53$x \\in (x_0-\\delta, x_0)$\u65f6\uff0c$f'(x) > 0$\uff0c\u5f53$x \\in (x_0, x_0+\\delta)$\u65f6\uff0c$f'(x) < 0$\uff0c\u5219$f(x)$\u5728$x_0$\u5904\u53d6\u5f97\u6781\u5927\u503c\u3002
\n\uff082\uff09\u82e5\u5f53$x \\in (x_0-\\delta, x_0)$\u65f6\uff0c$f'(x) < 0$\uff0c\u5f53$x \\in (x_0, x_0+\\delta)$\u65f6\uff0c$f'(x) > 0$\uff0c\u5219$f(x)$\u5728$x_0$\u5904\u53d6\u5f97\u6781\u5c0f\u503c\u3002
\n\uff083\uff09\u82e5\u5f53$x \\in U(x_0, \\delta)$\u65f6\uff0c$f'(x)$\u7b26\u53f7\u4fdd\u6301\u4e0d\u53d8\uff0c\u5219$f(x)$\u5728$x_0$\u5904\u65e0\u6781\u503c\u3002<\/p>\n

\n

\u6781\u503c\u53ef\u7591\u70b9\uff1a\u4e00\u9636\u5bfc\u6570\u4e3a0 \u6216\u5bfc\u6570\u4e0d\u5b58\u5728<\/p>\n<\/blockquote>\n

\u6781\u503c\u5b58\u5728\u7684\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6<\/h2>\n

\u8bbe\u51fd\u6570$f(x)$\u5728\u5b83\u7684\u9a7b\u70b9$x_0$\u5904\u4e8c\u9636\u53ef\u5bfc\uff0c\u5219<\/p>\n

\u82e5$f”(x_0)>0$\uff0c\u5219$x_0$\u4e3a\u6781\u5c0f\u503c\u70b9\uff1b
\n\u82e5$f”(x_0)<0$\uff0c\u5219$x_0$\u4e3a\u6781\u5927\u503c\u70b9\uff1b
\n\u82e5$f”(x_0)=0$\uff0c\u5219\u65e0\u6cd5\u5224\u65ad\u3002<\/p>\n

(1) \u6b64\u6cd5\u53ea\u9002\u7528\u4e8e\u9a7b\u70b9\uff0c\u4e0d\u80fd\u7528\u4e8e\u5224\u65ad\u4e0d\u53ef\u5bfc\u70b9
\n(2) \u5f53$f”(x_0)=0$\u65f6\uff0c\u5931\u6548\uff0c\u5982\uff1a$x^3$\u5728$x=0$\u5904<\/p>\n

\u6c42\u6781\u503c\u7684\u6b65\u9aa4<\/h2>\n

(1) \u786e\u5b9a\u51fd\u6570\u7684\u5b9a\u4e49\u57df
\n(2) \u6c42\u5bfc\u6570$f'(x)$
\n(3) \u6c42\u5b9a\u4e49\u57df\u5185\u90e8\u7684\u6781\u503c\u5acc\u7591\u70b9\uff08\u5373\u9a7b\u70b9\u6216\u4e00\u9636\u5bfc\u6570\u4e0d\u5b58\u5728\u7684\u70b9\uff09
\n(4) \u7528\u6781\u503c\u7684\u5224\u5b9a\u7b2c\u4e00\u6216\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6\uff08\u6ce8\u610f\u7b2c\u4e8c\u5145\u5206\u6761\u4ef6\u53ea\u80fd\u5224\u5b9a\u9a7b\u70b9\u7684\u60c5\u5f62\uff09<\/p>\n

\u4f8b\uff1a\u6c42\u51fa\u51fd\u6570$f(x)=x^3+3x^2-24x-20$\u7684\u6781\u503c<\/p>\n

\u89e3\uff1a$f'(x)=3x^2+6x-24=3(x+4)(x-2)$<\/p>\n

\u4ee4$f'(x)=0$\uff0c\u5f97\u9a7b\u70b9$x_1=-4,x_2=2$<\/p>\n

$\\because f”(-4)=-18<0$\uff0c\u6545\u6781\u5927\u503c$f(-4)=60$\uff1b$f”(2)=18>0$\uff0c\u6545\u6781\u5c0f\u503c$f(2)=-48$\u3002<\/p>\n

\u51fd\u6570\u7684\u6700\u5927\u503c\u3001\u6700\u5c0f\u503c\u95ee\u9898<\/h2>\n

\u6781\u503c\u662f\u5c40\u90e8\u6027\u7684\uff0c\u800c\u6700\u503c\u662f\u5168\u5c40\u6027\u7684\u3002\u82e5\u51fd\u6570$f(x)$\u5728$[a,b]$\u4e0a\u8fde\u7eed\uff0c\u5219$f(x)$\u5728$[a,b]$\u4e0a\u7684\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c\u5b58\u5728\u3002<\/p>\n

\u5177\u4f53\u6c42\u6cd5\uff1a<\/p>\n

(1) \u6c42\u51fa\u5b9a\u4e49\u57df\u5185\u90e8\u7684\u6781\u503c\u5acc\u7591\u70b9\uff08\u9a7b\u70b9\u548c\u4e0d\u53ef\u5bfc\u70b9\uff09$x_1,x_2,\u00b7\u00b7\u00b7,x_k$\uff0c\u5e76\u7b97\u51fa\u51fd\u6570\u503c$f(x_i)(i=1,2,\u00b7\u00b7\u00b7,k)$
\n(2) \u6c42\u51fa\u7aef\u70b9\u7684\u51fd\u6570\u503c$f(a),f(b)$
\n(3) \u6700\u5927\u503c$M=max{f(x_1),\u00b7\u00b7\u00b7,f(x_k),f(a),f(b)}$\uff0c\u6700\u5c0f\u503c$m=min{f(x_1),\u00b7\u00b7\u00b7,f(x_k),f(a),f(b)}$<\/p>\n

\u6c42\u89e3\u6280\u5de7\uff1a<\/p>\n

(1) \u5982\u679c$f(x)$\u5728$[a,b]$\u4e0a\u5355\u8c03\uff0c\u5219\u5b83\u7684\u6700\u503c\u5fc5\u5728\u7aef\u70b9\u5904\u6536\u5230
\n(2) \u5982\u679c$f(x)$\u5728$[a,b]$\u4e0a\u8fde\u7eed\uff0c\u4e14\u5728$(a,b)$\u5185\u53ef\u5bfc\uff0c\u4e14\u6709\u552f\u4e00\u9a7b\u70b9\uff0c\u5219\u82e5\u4e3a\u6781\u5c0f\u503c\u70b9\u5fc5\u4e3a\u6700\u5c0f\u503c\u70b9\uff0c\u82e5\u4e3a\u6781\u5927\u503c\u70b9\u5fc5\u4e3a\u6700\u5927\u503c\u70b9
\n(3) \u5982\u679c$f(x)$\u5728$[a,b]$\u4e0a\u6709\u6700\u5927(\u5c0f)\u503c\uff0c\u4e14\u6709\u552f\u4e00\u9a7b\u70b9\uff0c\u5219\u4e0d\u5fc5\u5224\u65ad\u6781\u5927\u8fd8\u662f\u6781\u5c0f\uff0c\u7acb\u5373\u53ef\u4ee5\u5224\u5b9a\u8be5\u9a7b\u70b9\u5373\u4e3a\u6700\u5927(\u5c0f)\u503c\u70b9<\/p>\n

\u4f8b\u9898\u89e3\u6790<\/h2>\n

\uff081\uff09\u6c42\u51fd\u6570$f(x)=(x-1)(x-2)^3(x-3)^3(x-4)$\u5728$(-\u221e,+\u221e)$\u4e0a\u7684\u6781\u503c\u6570<\/p>\n

\u7a7f\u7ebf\u6cd5\uff1a\u5947\u7a7f\u5076\u4e0d\u7a7f<\/p>\n

\u5f53$x < 1$\uff0c$2 < x < 4$\u4e14$x \\neq 3$\u65f6\uff0c$f(x) < 0$
\n\u5f53$1 < x < 2$\uff0c$x > 4$\u65f6\uff0c$f(x) > 0$<\/p>\n

\uff082\uff09\u5c06\u8fb9\u957f\u4e3a$a$\u7684\u6b63\u65b9\u5f62\u94c1\u76ae\uff0c\u56db\u89d2\u5404\u622a\u53bb\u76f8\u540c\u7684\u5c0f\u6b63\u65b9\u5f62\uff0c\u6298\u6210\u4e00\u4e2a\u65e0\u76d6\u65b9\u76d2\uff0c\u95ee\u5982\u4f55\u622a\u4f7f\u65b9\u76d2\u7684\u5bb9\u79ef\u6700\u5927\uff1f<\/p>\n

$V = x(a-2x)^2 \\, x \\in (0, \\frac{a}{2})$
\n$V’ = (2x-a)(6x-a)$
\n$V” = 24x-8a$<\/p>\n

\u552f\u4e00\u9a7b\u70b9$x=\\frac{a}{6}, \\, V”(\\frac{a}{6})=-4a<0$\uff0c\u53d6\u5f97\u6781\u5927\u503c<\/p>\n

\uff083\uff09\u8981\u505a\u4e00\u4e2a\u5bb9\u79ef\u4e3aV\u7684\u5706\u67f1\u5f62\u7f50\u5934\u7b52\uff0c\u600e\u6837\u8bbe\u8ba1\u624d\u80fd\u4f7f\u6240\u7528\u6750\u6599\u6700\u7701\uff1f<\/p>\n

\u5373\u8868\u9762\u79ef\u6700\u5c0f\uff0c\u8bbe\u5e95\u534a\u5f84\u4e3a$r$\uff0c\u9ad8\u4e3a$h$<\/p>\n

$V = \\pi r^2h \\Rightarrow h = \\frac{V}{\\pi r^2}$
\n$S = 2\\pi r^2 + 2\\pi rh = 2\\pi r^2 + 2\\pi r \\frac{V}{\\pi r^2} = 2\\pi r^2 + \\frac{2V}{r}$
\n$S’ = 4\\pi r – \\frac{2V}{r^2}$
\n$S” = 4\\pi + \\frac{2V}{r^3}$<\/p>\n

\u552f\u4e00\u9a7b\u70b9$x=\\sqrt[3]{\\frac{V}{2\\pi}}, \\, V”=12\\pi>0$\uff0c\u53d6\u5f97\u6781\u5c0f\u503c<\/p>\n","protected":false},"excerpt":{"rendered":"

\u51fd\u6570\u5355\u8c03\u6027 \u901a\u8fc7\u51fd\u6570\u7684\u5bfc\u6570\u7684\u503c\uff0c\u53ef\u4ee5\u5224\u65ad\u51fa\u51fd\u6570\u7684\u5355\u8c03\u6027\u3001\u9a7b\u70b9\u4ee5\u53ca\u6781\u503c\u70b9: \u82e5\u5bfc\u6570\u5927\u4e8e0\uff0c\u5219\u5355\u8c03\u9012\u589e\uff1b\u82e5\u5bfc\u6570\u5c0f\u4e8e […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2132","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2132","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2132"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2132\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}