ON THE MEAN VALUE THEOREM AND ROLLE’S THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES

Tóm tắt

The classical forms of Rolle’s Theorem and Lagrange's Mean Value Theorem for a differentiable, single-valued real function, are fundamental results in mathematical analysis, with many important applications. A natural and interesting question is to extend these theorems to the case of functions of several variables. In this paper, we present a version of the Mean Value Theorem for functions of several variables and provide an application of the classical Mean Value Theorem in functional equations. When extending the mean value theorem from the one-variable case to several variables, the function on an interval is replaced by a function defined on the closure of a domain, and the endpoint values are replaced by the values on the boundary of that domain. To prove our mean value theorem for differentiable functions of several real variables, we make use of a version of Rolle’s Theorem due to Alberto Fiorenza and Renato Fiorenza (2024).