关于中值证明题中辅助函数的构造

Study on Structure of Auxiliary Function in Mean Value Proof Questions

微分中值定理是沟通函数与导数之间的桥梁,是研究函数性质的有力工具。通常有洛尔中值定理,拉格朗日中值定理,柯西中值定理。其中又以洛尔中值定理为其基础,而拉格朗日中值定理,柯西中值定理以及其它有关中值证明题往往可以通过引入辅助函数,再借助洛尔中值定理得到圆满的解决。在这里辅助函数的构造就是证明的关键。本文针对含有一阶导数的中值证明题利用微分方程法进行辅助函数的构造。

The differential mean value theorem is the bridge between functions and the derivative, and a powerful tool for studying the properties of functions. Rolle Mean Value Theorem, Lagrange Mean Value Theorem and Cauchy Mean Value Theorem belong to Mean Value Theorem, among which Rolle Mean Value Theorem is its foundation while Lagrange Mean Value Theorem, Cauchy Mean Value Theorem and other relevant mean value theorem questions can be answered perfectly by auxiliary function and Rolle Mean Value Theorem. Here, the structure of auxiliary function is the key point of proof. To work out the mean value proof questions with first derivative, this thesis studies on the structure of auxiliary function through differential equation method.