微分方程在构造微分中值命题的辅助函数中的应用

Application of Differential Equations in Constructed Auxiliary Functions for Differential Propositions of Mean

众所周知,Lagrange定理、Cauchy定理[1]及其它许许多多微分中值命题的证明均借助于构造一个适当的辅助函数。然而,如何作辅助函数,如同作几何证明中的辅助线,需要较高的技巧,无一定法则可循,这给教学带来了难处。文[2]给出了一种辅助函数的“统一”构造法,只需按照一套固定的程序即可。本文利用简单微分方程的解构造出中值问题的辅助函数,从而得到寻求辅助函数的一种新方法。

As everybody knows, the proof of Lagrange's theorem, Cauchy's theorem and many propositions rekated with differention theorem of mean is by means of constructing a suitable auxiliary function. But how to construct an auxiliary function needs higher technique and has no rules to follow, as the same as making auxiliary line geometry proof problem. It has brought about difficulty for teaching work.In the paper[2], a unified construction method for auxiliary functions is given, only in accordance with regular procedure. This paper has constructed an auxiliary function for the differential propositions of mean by using solutions to simple differential equation. And then, we obtain a new method for exploring auxiliary functions.