如图f,g是复系数多项式，两者互素，关于导数与根的题
\u82e5\u4e00\u5b9e\u7cfb\u6570\u591a\u9879\u5f0f\u7684\u6839\u5168\u4e3a\u5b9e\u6839\uff0c\u5219\u4ed6\u7684\u5404\u9636\u5bfc\u6570\u7684\u6839\u5168\u4e3a\u5b9e\u6839\uff0c\u6c42\u8bc1\u660en\u6b21\u5b9e\u7cfb\u6570\u591a\u9879\u5f0ff(x)\u7684\u6839\u5168\u4e3a\u5b9e\u6570\uff0c\u5219\u53ef\u4ee5\u8868\u793a\u6210f(x)=a(x-x(1))(x-x(2))...(x-x(n)), a\u22600\uff0c \u5219
f'(x)=a(x-x(2))(x-x(3))...(x-x(n)) + a(x-x(1))(x-x(3))(x-x(4))...(x-x(n)) + ... + a(x-x(1))...(x-x(n-1))\uff0c
\u4e0d\u59a8\u8bbex(1) < x(2) < x(3) < ... < x(n)\uff0c\u5148\u4e0d\u8003\u8651\u91cd\u6839\u7684\u60c5\u51b5\uff0c\u5bb9\u6613\u9a8c\u8bc1
f'(x(1))f'(x(2)) < 0\uff0cf'(x(2))f'(x(3)) < 0\uff0c ... \uff0cf'(x(n-1))f'(x(n)) < 0\uff0c
\u6240\u4ee5f'(x)\u5728(x(1), x(2)), (x(2), x(3)),..., (x(n-1), x(n))\u8fd9n-1\u4e2a\u533a\u95f4\u5185\u5404\u6709\u4e00\u4e2a\u6839\uff0c\u6240\u4ee5f'(x)\u7684\u6839\u5168\u4e3a\u5b9e\u6570\u3002
\u5982\u679cf(x)\u6709\u91cd\u6839\uff0c\u5219\u53ef\u8bbex(1) \u2264 x(2) \u2264 x(3) \u2264... \u2264 x(n)\uff0c\u82e5x(k)<x(k+1)\uff0c\u7531\u4e0a\u9762\u7684\u8bc1\u660e\u53ef\u5f97
f'(x)\u5728(x(k), x(k+1))\u5185\u6709\u4e00\u4e2a\u5b9e\u6839\uff1b\u82e5x(k)=x(k+1)=...=x(k+m)\uff0cm\u22651\uff0c\u5bb9\u6613\u9a8c\u8bc1x(k)\u662ff'(x)\u7684m\u91cd\u6839\uff0c\u6240\u4ee5\u8fd9\u65f6\u4ecd\u7136\u53ef\u5f97f'(x)\u6709n-1\u4e2a\u5b9e\u6839\u3002
\u6240\u4ee5\u4e00\u9636\u5bfc\u6570f'(x)\u7684\u6839\u5168\u4e3a\u5b9e\u6570\uff0c\u4ee4g(x)=f'(x)\uff0c\u56e0\u4e3ag(x)\u7684\u6839\u5168\u4e3a\u5b9e\u6570\uff0c\u5219\u7531\u4e0a\u9762\u7684\u7ed3\u8bba\u53ef\u5f97
f''(x)=g'(x)\u7684\u6839\u5168\u4e3a\u5b9e\u6570\u3002\u5bf9\u591a\u9879\u5f0f\u7684\u6b21\u6570n\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u53ef\u5f97\uff0cf(x)\u7684\u5404\u9636\u5bfc\u6570\u7684\u6839\u5168\u4e3a\u5b9e\u6570\u3002
\u7b2c\u4e00\u4e2a\u9879\u7684\u7cfb\u6570\u662f-5\u6b21\u6570\u662f2a+1
\u7b2c\u4e8c\u4e2a\u9879\u7684\u7cfb\u6570\u662f\u8d1f\u56db\u5206\u4e4b\u4e00\uff0c\u6b21\u6570\u662f6
\u7b2c\u4e09\u4e2a\u9879\u7cfb\u6570\u662f\u4e09\u5206\u4e4b\u4e00\uff0c\u6b21\u6570\u662f5
\u6211\u8fd9\u91cc\u5b66\u7684\u86ee\u597d\u7684
\u671b\u7ed9\u5206\u6492
所以g'(t)=-f(t)f'(t)/g(t)，
f'^2(t)+g'^2(t) = f'^2(t) + f^2(t)f'^2(t)/g^2(t) = f'^2(t)/g^2(t) * (f^2(t)+g^2(t)) = 0
于是t是f'^2+g'^2的根
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