Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation

Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, Toulouse ( France )

^* jbhu@math.univ-toulouse.fr

Abstract

We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $\frac{f\left(a\right)-f\left(b\right)}{b-a}=f^{\text{'}}\left(c\right)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.