{"id":11070,"date":"2024-08-24T16:17:10","date_gmt":"2024-08-24T08:17:10","guid":{"rendered":"http:\/\/www.aiersheng.cc\/?p=11070"},"modified":"2024-08-24T16:17:10","modified_gmt":"2024-08-24T08:17:10","slug":"%e6%b5%99%e6%b1%9f%e7%9c%81%e4%b8%93%e5%8d%87%e6%9c%ac%e9%ab%98%e6%95%b0%e5%a4%87%e8%80%83%e5%bf%85%e7%9c%8b%ef%bc%9a%e5%8e%86%e5%b9%b4%e7%9c%9f%e9%a2%98%e6%a0%b8%e5%bf%83%e7%9f%a5%e8%af%86%e7%82%b9-3","status":"publish","type":"post","link":"https:\/\/www.aiersheng.cc\/?p=11070","title":{"rendered":"\u6d59\u6c5f\u7701\u4e13\u5347\u672c\u9ad8\u6570\u5907\u8003\u5fc5\u770b\uff1a\u5386\u5e74\u771f\u9898\u6838\u5fc3\u77e5\u8bc6\u70b9\u5168\u9762\u63d0\u70bc\u4e0e\u4f18\u5316\uff083\uff09"},"content":{"rendered":"

\u9488\u5bf9\u201c\u51fd\u6570\u56db\u6027\u201d\uff08\u6709\u754c\u6027\u3001\u5355\u8c03\u6027\u3001\u5947\u5076\u6027\u3001\u5468\u671f\u6027\uff09\u8fd9\u4e00\u77e5\u8bc6\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u5206\u522b\u901a\u8fc7\u4f8b\u9898\u6765\u52a0\u6df1\u7406\u89e3\u548c\u638c\u63e1\u3002\u4ee5\u4e0b\u662f\u5bf9\u6bcf\u79cd\u6027\u8d28\u7684\u7b80\u8981\u8bf4\u660e\u53ca\u4f8b\u9898\u793a\u4f8b\uff1a<\/p>\n

1. \u6709\u754c\u6027<\/h3>\n

\u8bf4\u660e<\/strong>\uff1a\u6709\u754c\u51fd\u6570\u6307\u7684\u662f\u5728\u67d0\u4e2a\u533a\u95f4\u5185\uff0c\u51fd\u6570\u7684\u503c\u57df\u662f\u6709\u9650\u7684\uff0c\u5373\u5b58\u5728\u4e24\u4e2a\u6570M\u548cm\uff08M>m\uff09\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u8be5\u533a\u95f4\u5185\u7684\u6240\u6709x\uff0c\u90fd\u6709m\u2264f(x)\u2264M\u3002<\/p>\n

\u4f8b\u9898<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f(x) = sinx\u5728\u533a\u95f4[0, \u03c0]\u4e0a\u662f\u5426\u6709\u754c\uff1f<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5728\u533a\u95f4[0, \u03c0]\u4e0a\uff0csinx\u7684\u503c\u57df\u4e3a[0, 1]\uff0c\u56e0\u6b64\u5b58\u5728M=1\u548cm=0\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709x\u2208[0, \u03c0]\uff0c\u90fd\u67090\u2264sinx\u22641\uff0c\u6240\u4ee5f(x) = sinx\u5728\u533a\u95f4[0, \u03c0]\u4e0a\u6709\u754c\u3002<\/p>\n

\u4f8b\u98982<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>x<\/span>2<\/span><\/span><\/span>+<\/span>1<\/span><\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5728\u5168\u4f53\u5b9e\u6570\u8303\u56f4R<\/span><\/span><\/span><\/span><\/span>\u4e0a\u662f\u5426\u6709\u754c\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5bf9\u4e8e\u4efb\u610fx<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span>\uff0c\u6709x<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\uff0c\u6240\u4ee5x<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>1<\/span>\u2265<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002\u8fdb\u800c\uff0c0<\/span><<\/span><\/span>x<\/span>2<\/span><\/span><\/span>+<\/span>1<\/span><\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64\uff0c\u5b58\u5728M<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u548cm<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\uff0c\u4f7f\u5f97m<\/span>\u2264<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span>\u2264<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span>\u5bf9\u6240\u6709x<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span>\u6210\u7acb\uff0c\u6240\u4ee5f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5728R<\/span><\/span><\/span><\/span><\/span>\u4e0a\u6709\u754c\u3002<\/p>\n

2. \u5355\u8c03\u6027<\/h3>\n

\u8bf4\u660e<\/strong>\uff1a\u5355\u8c03\u6027\u63cf\u8ff0\u7684\u662f\u51fd\u6570\u5728\u67d0\u4e2a\u533a\u95f4\u5185\u662f\u5355\u8c03\u589e\u52a0\u8fd8\u662f\u5355\u8c03\u51cf\u5c11\u3002<\/p>\n

\u4f8b\u98981<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f(x) = x^2\u5728\u533a\u95f4(0, +\u221e)\u4e0a\u7684\u5355\u8c03\u6027\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5bf9\u4e8e\u4efb\u610fx1, x2\u2208(0, +\u221e)\uff0c\u4e14x1<x2\uff0c\u6709f(x1) – f(x2) = x1^2 – x2^2 = (x1-x2)(x1+x2)\u3002\u7531\u4e8ex1, x2>0\uff0c\u6240\u4ee5x1+x2>0\uff0c\u4e14x1-x2<0\uff0c\u56e0\u6b64f(x1) – f(x2) < 0\uff0c\u5373f(x1) < f(x2)\u3002\u6240\u4ee5f(x) = x^2\u5728\u533a\u95f4(0, +\u221e)\u4e0a\u5355\u8c03\u589e\u52a0\u3002<\/p>\n

\u4f8b\u98982<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5728\u533a\u95f4(<\/span>0<\/span>,<\/span>+<\/span>\u221e<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4e0a\u7684\u5355\u8c03\u6027\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5bf9\u4e8e\u4efb\u610fx<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span>x<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span>(<\/span>0<\/span>,<\/span>+<\/span>\u221e<\/span>)<\/span><\/span><\/span><\/span><\/span>\uff0c\u4e14x<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><<\/span><\/span>x<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u6709x<\/span>1<\/span>\u200b<\/span><\/span><\/span><\/span>x<\/span>2<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002\u56e0\u4e3a\u5bf9\u6570\u51fd\u6570log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u662f\u589e\u51fd\u6570\uff0c\u6240\u4ee5log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>\u2212<\/span><\/span>log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>=<\/span><\/span>log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>x<\/span>1<\/span>\u200b<\/span><\/span><\/span><\/span>x<\/span>2<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\uff0c\u5373log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>><\/span><\/span>log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u6240\u4ee5f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>log2<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5728\u533a\u95f4(<\/span>0<\/span>,<\/span>+<\/span>\u221e<\/span>)<\/span><\/span><\/span><\/span><\/span>\u4e0a\u5355\u8c03\u589e\u52a0\u3002<\/p>\n

3. \u5947\u5076\u6027<\/h3>\n

\u8bf4\u660e<\/strong>\uff1a\u5947\u51fd\u6570\u6ee1\u8db3f(-x) = -f(x)\uff0c\u5076\u51fd\u6570\u6ee1\u8db3f(-x) = f(x)\u3002<\/p>\n

\u4f8b\u98981<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f(x) = x^3\u7684\u5947\u5076\u6027\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u51fd\u6570f(x) = x3\u7684\u5b9a\u4e49\u57df\u4e3a\u5168\u4f53\u5b9e\u6570R\uff0c\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\u3002\u8ba1\u7b97f(-x) = (-x)3 = -x3 = -f(x)\uff0c\u6240\u4ee5f(x) = x3\u662f\u5947\u51fd\u6570\u3002<\/p>\n

\u4f8b\u98982<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>\u2223<\/span>x<\/span>\u2223<\/span><\/span><\/span><\/span><\/span>\u7684\u5947\u5076\u6027\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u51fd\u6570f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>\u2223<\/span>x<\/span>\u2223<\/span><\/span><\/span><\/span><\/span>\u7684\u5b9a\u4e49\u57df\u4e3a\u5168\u4f53\u5b9e\u6570R<\/span><\/span><\/span><\/span><\/span>\uff0c\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\u3002\u5bf9\u4e8e\u4efb\u610fx<\/span>\u2208<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span>\uff0c\u6709f<\/span>(<\/span>\u2212<\/span>x<\/span>)<\/span>=<\/span><\/span>\u2223<\/span>\u2212<\/span><\/span>x<\/span>\u2223<\/span>=<\/span><\/span>\u2223<\/span>x<\/span>\u2223<\/span>=<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64\uff0cf<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>\u2223<\/span>x<\/span>\u2223<\/span><\/span><\/span><\/span><\/span>\u662f\u5076\u51fd\u6570\u3002<\/p>\n

4. \u5468\u671f\u6027<\/h3>\n

\u8bf4\u660e<\/strong>\uff1a\u5468\u671f\u51fd\u6570\u662f\u6307\u5b58\u5728\u4e00\u4e2a\u6b63\u6570T\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709x\uff0c\u90fd\u6709f(x+T) = f(x)\u3002T\u79f0\u4e3a\u51fd\u6570\u7684\u6700\u5c0f\u6b63\u5468\u671f\uff0c\u4f46\u5e76\u975e\u6240\u6709\u5468\u671f\u51fd\u6570\u90fd\u6709\u6700\u5c0f\u6b63\u5468\u671f\u3002<\/p>\n

\u4f8b\u98981<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f(x) = sinx\u7684\u5468\u671f\u6027\uff0c\u5e76\u6c42\u5176\u6700\u5c0f\u6b63\u5468\u671f\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5bf9\u4e8e\u4efb\u610fx\uff0c\u6709f(x+2\u03c0) = sin(x+2\u03c0) = sinx\uff08\u5229\u7528\u6b63\u5f26\u51fd\u6570\u7684\u5468\u671f\u6027\uff09\uff0c\u6240\u4ee5f(x) = sinx\u662f\u5468\u671f\u51fd\u6570\u3002\u5176\u4e2d\uff0c2\u03c0\u662fsinx\u7684\u4e00\u4e2a\u5468\u671f\uff0c\u4e14\u662f\u6700\u5c0f\u7684\u6b63\u5468\u671f\uff0c\u56e0\u6b64sinx\u7684\u6700\u5c0f\u6b63\u5468\u671f\u4e3a2\u03c0\u3002<\/p>\n

\u4f8b\u98982<\/strong>\uff1a\u5224\u65ad\u51fd\u6570f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>cos<\/span>(<\/span>2<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u5468\u671f\u6027\uff0c\u5e76\u6c42\u5176\u6700\u5c0f\u6b63\u5468\u671f\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5bf9\u4e8e\u4efb\u610fx<\/span><\/span><\/span><\/span><\/span>\uff0c\u6709f<\/span>(<\/span>x<\/span>+<\/span><\/span>\u03c0<\/span>)<\/span>=<\/span><\/span>cos<\/span>(<\/span>2<\/span>(<\/span>x<\/span>+<\/span><\/span>\u03c0<\/span>))<\/span>=<\/span><\/span>cos<\/span>(<\/span>2<\/span>x<\/span>+<\/span><\/span>2<\/span>\u03c0<\/span>)<\/span>=<\/span><\/span>cos<\/span>(<\/span>2<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\uff08\u5229\u7528\u4f59\u5f26\u51fd\u6570\u7684\u5468\u671f\u6027\uff09\u3002\u56e0\u6b64\uff0c\u03c0<\/span><\/span><\/span><\/span><\/span>\u662ff<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>cos<\/span>(<\/span>2<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u4e00\u4e2a\u5468\u671f\u3002\u53c8\u56e0\u4e3a2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u4e0d\u662ff<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u66f4\u5c0f\u6b63\u5468\u671f\uff08\u53ef\u4ee5\u9a8c\u8bc1f<\/span>(<\/span>x<\/span>+<\/span><\/span>2<\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\uff09\uff0c\u6240\u4ee5\u03c0<\/span><\/span><\/span><\/span><\/span>\u662ff<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>cos<\/span>(<\/span>2<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u5c0f\u6b63\u5468\u671f\u3002<\/p>\n

\u4f8b\u98983<\/strong>\uff08\u6d89\u53ca\u975e\u4f20\u7edf\u5468\u671f\u51fd\u6570\uff09\uff1a\u5224\u65ad\u51fd\u6570f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u5468\u671f\u6027\uff0c\u5e76\u6c42\u5176\u6700\u5c0f\u6b63\u5468\u671f\u3002<\/p>\n

\u89e3\u7b54<\/strong>\uff1a\u5229\u7528\u4e09\u89d2\u6052\u7b49\u5f0fsin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>2<\/span><\/span>1\u2212<\/span>c<\/span>o<\/span>s<\/span><\/span>(<\/span>2x<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u8f6c\u5316\u4e3a\u4e0e\u4f59\u5f26\u51fd\u6570\u76f8\u5173\u7684\u5f62\u5f0f\u3002\u7531\u4e8ecos<\/span>(<\/span>2<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u5468\u671f\u4e3a\u03c0<\/span><\/span><\/span><\/span><\/span>\uff0c\u90a3\u4e48sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u5468\u671f\u4e5f\u662f2<\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff08\u56e0\u4e3a2<\/span>x<\/span><\/span><\/span><\/span><\/span>\u4e2d\u7684\u7cfb\u65702\u4f7f\u5f97\u5468\u671f\u51cf\u534a\uff09\u3002\u9a8c\u8bc1\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u6709f<\/span>(<\/span>x<\/span>+<\/span><\/span>2<\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>+<\/span><\/span>2<\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span>=<\/span><\/span>cos2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>1<\/span>\u2212<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span>\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\uff0c\u4f46f<\/span>(<\/span>x<\/span>+<\/span><\/span>\u03c0<\/span>)<\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>+<\/span><\/span>\u03c0<\/span>)<\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\uff0c\u4e14\u03c0<\/span><\/span><\/span><\/span><\/span>\u4e0d\u662f\u66f4\u5c0f\u7684\u6b63\u5468\u671f\uff0c\u6240\u4ee5f<\/span>(<\/span>x<\/span>)<\/span>=<\/span><\/span>sin2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u5c0f\u6b63\u5468\u671f\u4e3a2<\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u901a\u8fc7\u4ee5\u4e0a\u4f8b\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u7406\u89e3\u548c\u638c\u63e1\u51fd\u6570\u7684\u6709\u754c\u6027\u3001\u5355\u8c03\u6027\u3001\u5947\u5076\u6027\u548c\u5468\u671f\u6027\u8fd9\u56db\u79cd\u57fa\u672c\u6027\u8d28\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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