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We study the sensitivity, essentially the differentiability, of the so-called “intermediate point”  c  in the classical mean value theorem $ \frac{f(a)-f(b)}{b-a}={f}^{\prime}(c)$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.

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