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Means and the mean value theorem
International Journal of Mathematical Education in Science and Technology, 2009Let 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 +2 more
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Ukrainian Mathematical Journal, 2014
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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, 2020Let 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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A Restricted Mean Value Theorem
Journal of the London Mathematical Society, 1969For 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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On a Certain Mean Value Theorem
Moscow University Mathematics Bulletin, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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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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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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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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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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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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(1969). On Mean Value Theorems. The American Mathematical Monthly: Vol. 76, No. 1, pp. 70-73.
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1991
Let $$ f\left( \lambda \right) = {{\alpha }_{k}}{{\lambda }^{k}} + \cdots + {{\alpha }_{1}}\lambda $$ be a polynomial of k-th degree with coefficients in J, where \({{\alpha }_{i}} \in M\left( {O({{T}^{{k - i}}})} \right), 1 \leqslant i \leqslant k\) Let $$ s\left( {f\left( \lambda \right)} \right),\xi ,{\text{T}} = s\left( {f,{\text{T ...
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Let $$ f\left( \lambda \right) = {{\alpha }_{k}}{{\lambda }^{k}} + \cdots + {{\alpha }_{1}}\lambda $$ be a polynomial of k-th degree with coefficients in J, where \({{\alpha }_{i}} \in M\left( {O({{T}^{{k - i}}})} \right), 1 \leqslant i \leqslant k\) Let $$ s\left( {f\left( \lambda \right)} \right),\xi ,{\text{T}} = s\left( {f,{\text{T ...
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Mean-Value Theorems in Arithmetic Semigroups
Acta Mathematica Hungarica, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Lucht, L. G., Reifenrath, K.
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