Abstract
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.
Highlights
The basic mean value theorem (MVT for short), called Lagrange’s MVT, is certainly one of the best known results in Calculus
The MVT theorem is mainly an existence result, it does not say more on this “intermediate point c”..., especially if it is unique or not
With the help of the generating function x > 0 → f (x) = exp x, one gets at c(a, b) = ln eb −ea b−a
Summary
The basic mean value theorem (MVT for short), called Lagrange’s MVT, is certainly one of the best known results in Calculus. With the help of the generating function x > 0 → f (x) = exp x, one gets at c(a, b) = ln eb −ea b−a. According to what has been seen above, the (positively homogeneous) harmonic mean could only come from a function of the type f : x > 0 → f (x) = xp (where p = 0 and p = 1). Is c a differentiable function of a and b? If so, what is the gradient vector of c at any point (d, d) lying on the critical diagonal line of R × R? We answer both questions, delineating the appropriate (minimal) general assumptions on f for that
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