Geodesic
Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation
Jean-Baptiste Hiriart-Urruty1
1 Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, Toulouse (France)
ESAIM. Proceedings, Tome 71 (2021), pp. 114-120.

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We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem 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.
DOI : 10.1051/proc/202171114

Jean-Baptiste Hiriart-Urruty 1

1 Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, Toulouse (France) 
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Jean-Baptiste Hiriart-Urruty. Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation. ESAIM. Proceedings, Tome 71 (2021), pp. 114-120. doi : 10.1051/proc/202171114. http://geodesic.mathdoc.fr/articles/10.1051/proc/202171114/

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