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)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when...
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| Format: | Article |
| Language: | English |
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EDP Sciences
2021-08-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | https://www.esaim-proc.org/articles/proc/pdf/2021/02/proc2107110.pdf |
| Summary: | We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem fa-f(b)b-a=f'(c)$ \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. |
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| ISSN: | 2267-3059 |