与Cauchy微分中值定理相关的几个问题

Problems Related to Cauchy's Differential Mean Value Theorem

根据Cauchy微分中值定理表达式的结构引入辅助函数F(x,c)=f(x)-f(c)/g(x)-g(c)(a<c<b),通过讨论其可导性,得到相关的几个不等式,由此得出Cauchy微分中值定理存在唯一"中值点"的一个条件,并给出其逆定理的一个较弱表述.

Inspired by Cauchy＇s differential mean value theorem, we discuss the differentiability ofthe function F（x,c）=f（x）-f（c）/g（x）-g（c） （a〈c〈b）, and obtain several inequalities related. Using theseinequalities, we obtain a sufficient condition for the uniqueness of the ＂mean value point＂ in Cauchy^s differential mean value theorem, and a weak form for its inverse theorem.