fx在-a,a上连续，有二阶连续导，f0=0，证明:存在一点ξ使得a³fξ=3∮-a 设fx在a，b上连续在a，b内二阶可导,且有fa=fc=fb...
\u8bbefx\u5728[0\uff0ca]\u4e0a\u8fde\u7eed\u5728(0\uff0ca)\u5185\u53ef\u5bfc\u4e14fa=0\u8bc1\u660e\u5b58\u5728\u4e00\u70b9\u03be\u5c5e\u4e8e(0\uff0ca)\u4f7ff\u03be+\u03bef'\u03be=\u8bbe g(x)=f(x)*x^3
\u5219\u6709\uff1ag'(x)=f(x)*3*x^2+f'(x)*x^3
\u56e0\u4e3a\uff1ag(0)=g(a)=0
\u6839\u636e\u4e2d\u503c\u5b9a\u7406\uff0c\u5728(0,a)\u4e2d\u5b58\u5728\u03be\u4f7f\u5f97g'(\u03be)=0
\u5373\uff1af(\u03be)*3*\u03be^2+f'(\u03be)*\u03be^3=0
\u6240\u4ee5\uff1af(\u03be)*3+f'(\u03be)*\u03be=0
\u6269\u5c55\u8d44\u6599\uff1a\u51fd\u6570\u4e0e\u5176\u5bfc\u6570\u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u51fd\u6570\uff1b\u800c\u5bfc\u6570\u53ea\u662f\u53cd\u6620\u51fd\u6570\u5728\u4e00\u70b9\u7684\u5c40\u90e8\u7279\u5f81\uff1b\u5982\u679c\u8981\u4e86\u89e3\u51fd\u6570\u5728\u5176\u5b9a\u4e49\u57df\u4e0a\u7684\u6574\u4f53\u6027\u6001\uff0c\u5c31\u9700\u8981\u5728\u5bfc\u6570\u53ca\u51fd\u6570\u95f4\u5efa\u7acb\u8d77\u8054\u7cfb\uff0c\u5fae\u5206\u4e2d\u503c\u5b9a\u7406\u5c31\u662f\u8fd9\u79cd\u4f5c\u7528\u3002
\u5fae\u5206\u4e2d\u503c\u5b9a\u7406\uff0c\u5305\u62ec\u7f57\u5c14\u5b9a\u7406\u3001\u62c9\u683c\u6717\u65e5\u5b9a\u7406\u3001\u67ef\u897f\u5b9a\u7406\u3001\u6cf0\u52d2\u5b9a\u7406\u3002\u662f\u6c9f\u901a\u5bfc\u6570\u503c\u4e0e\u51fd\u6570\u503c\u4e4b\u95f4\u7684\u6865\u6881\uff0c\u662f\u5229\u7528\u5bfc\u6570\u7684\u5c40\u90e8\u6027\u8d28\u63a8\u65ad\u51fd\u6570\u7684\u6574\u4f53\u6027\u8d28\u7684\u5de5\u5177\u3002
\u4ee5\u7f57\u5c14\u5b9a\u7406\u3001\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\u548c\u67ef\u897f\u4e2d\u503c\u5b9a\u7406\u7ec4\u6210\u7684\u4e00\u7ec4\u4e2d\u503c\u5b9a\u7406\u662f\u4e00\u6574\u4e2a\u5fae\u5206\u5b66\u7684\u7406\u8bba\u57fa\u7840\u3002
\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\uff0c\u5efa\u7acb\u4e86\u51fd\u6570\u503c\u4e0e\u5bfc\u6570\u503c\u4e4b\u95f4\u7684\u5b9a\u91cf\u8054\u7cfb\uff0c\u56e0\u800c\u53ef\u7528\u4e2d\u503c\u5b9a\u7406\u901a\u8fc7\u5bfc\u6570\u53bb\u7814\u7a76\u51fd\u6570\u7684\u6027\u6001\uff1b
\u4e2d\u503c\u5b9a\u7406\u7684\u4e3b\u8981\u4f5c\u7528\u5728\u4e8e\u7406\u8bba\u5206\u6790\u548c\u8bc1\u660e\uff1b\u540c\u65f6\u7531\u67ef\u897f\u4e2d\u503c\u5b9a\u7406\u8fd8\u53ef\u5bfc\u51fa\u4e00\u4e2a\u6c42\u6781\u9650\u7684\u6d1b\u5fc5\u8fbe\u6cd5\u5219\u3002
\u4e2d\u503c\u5b9a\u7406\u7684\u5e94\u7528\u4e3b\u8981\u662f\u4ee5\u4e2d\u503c\u5b9a\u7406\u4e3a\u57fa\u7840\uff0c\u5e94\u7528\u5bfc\u6570\u5224\u65ad\u51fd\u6570\u4e0a\u5347\uff0c\u4e0b\u964d\uff0c\u53d6\u6781\u503c\uff0c\u51f9\u5f62\uff0c\u51f8\u5f62\u548c\u62d0\u70b9\u7b49\u9879\u7684\u91cd\u8981\u6027\u6001\u3002
\u4ece\u800c\u80fd\u628a\u63e1\u4f4f\u51fd\u6570\u56fe\u8c61\u7684\u5404\u79cd\u51e0\u4f55\u7279\u5f81\u3002\u5728\u6781\u503c\u95ee\u9898\u4e0a\u4e5f\u6709\u91cd\u8981\u7684\u5b9e\u9645\u5e94\u7528\u3002
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1-\u4e2d\u503c\u5b9a\u7406
\u8bc1\uff1a
f(x)\u5728[a\uff0cc]\u4e0a\u8fde\u7eed\uff0c\u4e14\u5728(a\uff0cc)\u5185\u53ef\u5bfc
f(a)=f(c)
\u7531\u7f57\u5c14\u4e2d\u503c\u5b9a\u7406\u5f97\uff1a\u5728(a\uff0cc)\u5185\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\u03b7₁\uff0c\u4f7f\u5f97
f'(\u03b7₁)=[f(c)-f(a)]/(c-a)=0
\u540c\u7406\uff0c\u5728(c\uff0cb)\u5185\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\u03b7₂\uff0c\u4f7f\u5f97
f'(\u03b7₂)=[f(b)-f(c)]/(b-c)=0
\u7531\u7f57\u5c14\u4e2d\u503c\u5b9a\u7406\u5f97\uff1a\u5728(\u03b7₁\uff0c\u03b7₂)\u5185\uff0c\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\u03be\uff0c\u4f7f\u5f97
f''(\u03be)=[f(\u03b7₂)-f(\u03b7₁)]/(\u03b7₂-\u03b7₁)=0
\u03b7₁\u2208(a\uff0cc)\uff0c\u03b7₂\u2208(c\uff0cb)
\u56e0\u6b64\uff0c\u5728(a\uff0cb)\u5185\uff0c\u5b58\u5728\u03be\u4f7f\u5f97f''(\u03be)=0
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