拉格朗日中值定理的应用及推广
The Application and Extension of Lagrange’s Mean Value Theorem

在教学过程中，我们发现拉格朗日中值定理是学生学习微积分的巨大障碍，这是因为拉格朗日中值定理是微分中值定理的核心内容，是研究函数与导数之间联系的理论工具，在微积分学中起着至关重要的作用，应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用，以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。
During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.