Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation
We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem fa-f(b)b-a=f'(c)$ \frac{f(a)-f(b)}{b-a}={f}^{\prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when...
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EDP Sciences
2021-08-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | https://www.esaim-proc.org/articles/proc/pdf/2021/02/proc2107110.pdf |
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doaj-d342b9404c7b4777b0f63b6528b68b702021-09-02T09:29:22ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592021-08-017111412010.1051/proc/202171114proc2107110Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformationHiriart-Urruty Jean-Baptiste0Institut de Mathématiques, Université Paul SabatierWe study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem fa-f(b)b-a=f'(c)$ \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.https://www.esaim-proc.org/articles/proc/pdf/2021/02/proc2107110.pdf |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Hiriart-Urruty Jean-Baptiste |
| spellingShingle |
Hiriart-Urruty Jean-Baptiste Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation ESAIM: Proceedings and Surveys |
| author_facet |
Hiriart-Urruty Jean-Baptiste |
| author_sort |
Hiriart-Urruty Jean-Baptiste |
| title |
Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation |
| title_short |
Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation |
| title_full |
Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation |
| title_fullStr |
Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation |
| title_full_unstemmed |
Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation |
| title_sort |
sensitivity of the “intermediate point” in the mean value theorem: an approach via the legendre-fenchel transformation |
| publisher |
EDP Sciences |
| series |
ESAIM: Proceedings and Surveys |
| issn |
2267-3059 |
| publishDate |
2021-08-01 |
| description |
We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem fa-f(b)b-a=f'(c)$ \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. |
| url |
https://www.esaim-proc.org/articles/proc/pdf/2021/02/proc2107110.pdf |
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AT hiriarturrutyjeanbaptiste sensitivityoftheintermediatepointinthemeanvaluetheoremanapproachviathelegendrefencheltransformation |
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