Results 231 to 240 of about 745,915 (284)

On the mean value theorem

Optimization, 1988
Mean value theorems for nonsmooth functions are presented. Two versions are given, both using the contingent derivative. In. the first one a tangential convexity condition is used. In the second one no convexity assumption is made but the estimate. involves the contingent derivative df (x, b − a) of f at points arbitrarily close to the segment [a, b ...
exaly   +2 more sources

ON THE FRACTIONAL MEAN-VALUE THEOREM

International Journal of Bifurcation and Chaos, 2012
In this paper, we derive a fractional mean-value theorem both in the sense of Riemann–Liouville and in the sense of Caputo. This new formulation is more general than the generalized Taylor's formula of Kolwankar and the fractional mean-value theorem in the sense of Riemann–Liouville developed by Trujillo.
Peng Guo, Changpin Li, Guanrong Chen
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Means and the mean value theorem

International Journal of Mathematical Education in Science and Technology, 2009
Let I be a real interval. We call a continuous function μ : I × I → ℝ a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I → ℝ be a differentiable and strictly convex or strictly concave function. If a, b ∈ I with a ≠ b, then there exists a unique number ξ between a and b such that f(b) − f(a) = f ′(ξ)(b − a).
Jorma K. Merikoski, Markku Halmetoja, Timo Tossavainen   +2 more
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Mean-Value Theorem

Ukrainian Mathematical Journal, 2014
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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A New Proof of the Equivalence of the Cauchy Mean Value Theorem and the Mean Value Theorem

The American Mathematical Monthly, 2020
Let f,g:[a,b]→R be differentiable in (a, b) and continuous in [a,b] . The Cauchy mean value theorem states that, if g′(x)≠0 in (a, b), there is a number c∈(a,b) such that (1) f(b)−f(a)g(b)−g(a)=f′(...
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On a Certain Mean Value Theorem

Moscow University Mathematics Bulletin, 2019
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A Restricted Mean Value Theorem

Journal of the London Mathematical Society, 1969
For each prime \(p\) let \(f(p)\) denote the least integer solution \(n\) to the Legendre character conditions \[ \left(\frac{n+a_j}{p}\right) = \varepsilon_j, \quad (j=1,\ldots,k). \] Elliott shows that there exist positive constants \(\alpha\), \(A\) so that \[ \left(\sum_{p\le x} \min(f(p),x^\alpha)\right)/\pi(x) \rightarrow A\quad\text{as }x\to ...
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A Mean Value Theorem

The American Mathematical Monthly, 1999
Several theorems go by this name. The present note adds to the assortment an unusual variant (Theorem 1), which involves the shape of the underlying region in an interesting way. We work in Euclidean spaces, although Lemma 2 and the second inequality of Lemma 3 carry over to general Riemannian manifolds.
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On Mean Value Theorems

The American Mathematical Monthly, 1969
(1969). On Mean Value Theorems. The American Mathematical Monthly: Vol. 76, No. 1, pp. 70-73.
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Mean value theorem

2002
The derivative of a function f at a point ξ $$f'\left( \xi \right) = \mathop {\lim }\limits_{\Delta x \to 0} {\rm{ }}{{f\left( {\xi + \Delta x} \right) - f\left( \xi \right)} \over {\Delta x}},$$ is the slope of the line tangent to the graph of f at the point P = (ξ ,f (ξ)).
Adi Ben-Israel, Robert Gilbert
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