设fx在[0,1]上连续在(0,1)内可导且f（0）=f(1)=0 Fx在［0，1］上连续，在(0，1)内可导，且f0=0，f1...
\u8bbefx\u5728[0,1]\u4e0a\u8fde\u7eed\u5728(0,1)\u5185\u53ef\u5bfc\u4e14f(1)=0\u8bc1\u660e\u5b58\u5728\u4e00\u70b9\u03be\u5c5e\u4e8e(0,1)\u4f7f2f(\u03be)+\u03bef'(\u03be)=0\u8bc1\u660e\uff1a\u4ee4g(x)=x^2\uff0cG(x)=g(x)*f(x)\u3002
\u56e0\u4e3af(x)\u5728[0,1]\u4e0a\u8fde\u7eed\u5728(0,1)\u5185\u53ef\u5bfc\uff0c\u4e14g(x)\u5728[0,1]\u4e0a\u8fde\u7eed\u5728(0,1)\u5185\u53ef\u5bfc\uff0c
\u90a3\u4e48G(x)=g(x)*f(x)\u5728[0,1]\u4e0a\u8fde\u7eed\u5728(0,1)\u5185\u53ef\u5bfc\u3002
\u4e14G(x)'=(g(x)*f(x))'=(x^2*f(x))'
=x^2f'(x)+2xf(x)
\u800cG(0)=g(0)*f(0)=0*f(0)=0
G(1)=g(1)*f(1)=g(1)*0=0\uff0c
\u5373G(0)=G(1)\uff0c
\u90a3\u4e48\u5728(0,1)\u5185\u5b58\u5728\u4e00\u70b9\u03be\uff0c\u4f7fG(x)'=0
\u5373G(\u03be)'=0
\u03be^2f'(\u03be)+2\u03bef(\u03be)=0\uff0c\u53c8\u03be\u22600\uff0c\u5219\u03bef'(\u03be)+2f(\u03be)=0
\u6269\u5c55\u8d44\u6599\uff1a
1\u3001\u7f57\u5c14\u4e2d\u503c\u5b9a\u7406\u7684\u51e0\u4f55\u610f\u4e49
\u82e5\u8fde\u7eed\u66f2\u7ebfy=f(x) \u5728\u533a\u95f4 [a,b] \u4e0a\u6240\u5bf9\u5e94\u7684\u5f27\u6bb5 AB\uff0c\u9664\u7aef\u70b9\u5916\u5904\u5904\u5177\u6709\u4e0d\u5782\u76f4\u4e8e x \u8f74\u7684\u5207\u7ebf\uff0c\u4e14\u5728\u5f27\u7684\u4e24\u4e2a\u7aef\u70b9 A,B \u5904\u7684\u7eb5\u5750\u6807\u76f8\u7b49\uff0c\u5219\u5728\u5f27 AB \u4e0a\u81f3\u5c11\u6709\u4e00\u70b9 C\uff0c\u4f7f\u66f2\u7ebf\u5728C\u70b9\u5904\u7684\u5207\u7ebf\u5e73\u884c\u4e8e x \u8f74\u3002
2\u3001\u7f57\u5c14\u4e2d\u503c\u5b9a\u7406\u7684\u8bc1\u660e
\uff081\uff09\u82e5\u51fd\u6570f(x)\u5728\u533a\u95f4(a,b)\u4e0a\u8fde\u7eed\u4e14\u53ef\u5bfc\uff0c\u5e76\u6709lim(x\u2192a+0)f(x)=lim(x\u2192b-0)f(x)=A\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u03be\u2208(a,b)\uff0c\u4f7f\u5f97f(\u03be)'=0\u3002
\uff082\uff09\u82e5\u51fd\u6570f(x)\u5728\u533a\u95f4(a,b)\u4e0a\u8fde\u7eed\u4e14\u53ef\u5bfc\uff0c\u5e76\u6709lim(x\u2192a+0)f(x)=lim(x\u2192b-0)f(x)=+\u221e(-\u221e)\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u03be\u2208(a,b)\uff0c\u4f7f\u5f97f(\u03be)'=0\u3002
\uff083\uff09\u82e5\u51fd\u6570f(x)\u5728\u533a\u95f4(-\u221e,+\u221e)\u4e0a\u8fde\u7eed\u4e14\u53ef\u5bfc\uff0c\u5e76\u6709lim(x\u2192a+0)f(x)=lim(x\u2192b-0)f(x)=A\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u03be\u2208(-\u221e,+\u221e)\uff0c\u4f7f\u5f97f(\u03be)'=0\u3002
\uff084\uff09\u82e5\u51fd\u6570f(x)\u5728\u533a\u95f4(-\u221e,+\u221e)\u4e0a\u8fde\u7eed\u4e14\u53ef\u5bfc\uff0c\u5e76\u6709lim(x\u2192a+0)f(x)=lim(x\u2192b-0)f(x)=+\u221e(-\u221e)\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u03be\u2208(-\u221e,+\u221e)\uff0c\u4f7f\u5f97f(\u03be)'=0\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u7f57\u5c14\u4e2d\u503c\u5b9a\u7406
^\u4ee4g(x)=x^3*f(x)\uff0c\u5219g(x)\u5728[0,1]\u4e0a\u8fde\u7eed\uff0c\u5728(0,1)\u5185\u53ef\u5bfc
\u56e0\u4e3ag(0)=0\uff0cg(1)=f(1)=0\uff0c\u6240\u4ee5\u6839\u636e\u7f57\u5c14\u5b9a\u7406
\u5b58\u5728\u03be\u2208(0,1)\uff0c\u4f7f\u5f97g'(\u03be)=0
3\u03be^2*f(\u03be)+\u03be^3*f'(\u03be)=0
3f(\u03be)+\u03bef'(\u03be)=0
\u4e3b\u8981\u4f18\u52bf\uff1a
\u5219\u56e0\u4e3a f(a)=f(b) \u4f7f\u5f97\u6700\u5927\u503c M \u4e0e\u6700\u5c0f\u503c m \u81f3\u5c11\u6709\u4e00\u4e2a\u5728 (a,b) \u5185\u67d0\u70b9\u03be\u5904\u53d6\u5f97\uff0c\u4ece\u800c\u03be\u662ff(x)\u7684\u6781\u503c\u70b9\uff0c\u53c8\u6761\u4ef6 f(x) \u5728\u5f00\u533a\u95f4 (a,b) \u5185\u53ef\u5bfc\u5f97\uff0cf(x) \u5728 \u03be \u5904\u53d6\u5f97\u6781\u503c\uff0c\u7531\u8d39\u9a6c\u5f15\u7406\uff0c\u53ef\u5bfc\u7684\u6781\u503c\u70b9\u4e00\u5b9a\u662f\u9a7b\u70b9\uff0c\u63a8\u77e5\uff1af'(\u03be)=0\u3002
F'(x)=2xf(x)+x²f'(x)。
所以，2ξf(ξ)+ξ²f'(ξ)=0，所以2f(ξ)+ξf'(ξ)=0。
绛旓細绛夊紡宸﹁竟锛岀Н鍒嗕腑鍊煎畾鐞嗭細3*f(尉)*(1-2/3)=f(尉)=f(0) (0<尉<1)鍥犳锛锛0锛1锛夊唴鑷冲皯涓涓瀬鍊肩偣锛屽嵆瀛樺湪尉锛0锛1锛夛紝浣縡 '(尉)=0 寰楄瘉
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绛旓細瀹屽叏鍙互锛屽彲瀵煎繀杩炵画
