微分中值定理证明方法的思考

Pondering on Methods to Prove the Differential Mean Value Theorem

高等数学的教材是以罗尔定理为基础,通过引进适当满足罗尔定理的辅助函数去证明拉格朗日中值定理和柯西中值定理。本文将讨论如何构造辅助函数去证明拉格朗日中值定理和柯西中值定理。此外,本文还给出了证明微分中值定理的另外一种方法:辅助定理法。

In advanced mathematics textbooks, it is based on the Rolle Theorem, by introducing an appropriate auxiliary function that satisfies the Rolle Theorem to prove the Lagrange Mean Value Theorem and the Cauchy Mean Value Theorem. This paper will discuss how to construct an auxiliary function to prove Lagrange Mean Value Theorem and Cauchy Mean Value Theorem.In addition, the paper also gives another method to prove the differential mean value theorem： Lemma method.