A Simple Auxiliary Function for the Mean Value Theorem

Abstract

This is certainly an improvement because an additional hypothesis is justifiable only if it delivers a stronger conclusion or an easier proof. I know of no proof of Rolle's theorem that makes use of the vanishing of / at the end points. The main (only?) purpose of Rolle's theorem is to serve as a lemma for the more general mean value theorem. Whether (1) or (1') is given in Rolle's theorem, the mean value theorem is usually proved by first introducing the auxiliary function