The Mean Value Theorem and It’s Application

Abstract

The Mean Value Theorem (MVT) is the central theorem of differential Calculus. It’s the major tool to research on function, also bridging the gap between function and derivative. There are lot of researchers focus on it from ancient times until now. This article introduces details about two mean theorems include the Rolle’s Theorem and Lagrange Theorem. This paper uses the sample question and refutation to explain the conditions of the theorems, thus addressing the questions that many students will ask when learning definitions of these theorems. The article later examines the application of the MVT. The proof of the unique existence of roots and the inequality is included. The Mean Value Theorem reveals the link between the macroscopic, overall properties for a function on an interval and the microscopic, localized properties of the function at a point. The significance of the application for the mean value theorem is profound, not only promoting the development of mathematical analysis theory, but also playing an important role in multiple fields