Functional Equations and the Cauchy Mean Value Theorem

TL;DR

This paper characterizes pairs of smooth functions where the Cauchy Mean Value Theorem's mean value point is well-defined within the interval, providing insights into a question posed by Sahoo and Riedel.

Contribution

It offers a characterization of function pairs with well-positioned mean value points in the Cauchy Mean Value Theorem, addressing a specific open question.

Findings

Identifies conditions for well-positioned mean value points

Provides a partial answer to Sahoo and Riedel's question

Enhances understanding of the Cauchy Mean Value Theorem

Abstract

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.