对一个定理使用的商榷

A Brief Discussion on Application of a Theorem

文[1]中的定理3给出了结论"(ii)满足(1)式的中间点ξ=ξ(x)是x的可导函数,其导数为ξ′(x)=f′(x)g′(ξ(x)-f′(ξ(x))g′(x))(x-a)[f″(ξ(x))g′(ξ(x))-f′(ξ(x))g″(ξ(x))]"。文[1]在推导此等式时用到了柯西中值定理,本文指出在推导过程中使用柯西中值定理存在的问题,并给出例子对存在的问题作出详细的说明。

A conclusion reached by Theorem 3 in the essay（1） goes like this： ＂In ξ=ξ（x）,x is differentiable function,if a mean value in equation（1） is satisfied by（ii）,and therefore its derivative must be ξ/（x）=f/（x）g/（ξ（x）-f/（ξ（x））g/（x））/（x-a）[f//（ξ（x））g/（ξ（x））-f/（ξ（x））g//（ξ（x））]＂ This equation is derived by using Cauchy mean value theorem in the essay（1）.This paper puts forward some different opinions about the use of Cauchy mean value theorem in the essay（1）.