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