Mean Value Theorem

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mean value theorem

[′mēn ′val·yü ‚thir·əm]
(mathematics)
The proposition that, if a function ƒ (x) is continuous on the closed interval [a,b ] and differentiable on the open interval (a,b), then there exists x0, a <>x0<>b, such that ƒ(b) - ƒ(a) = (b-a)ƒ′(x0). Also known as first law of the mean; Lagrange's formula; law of the mean.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Mean Value Theorem

one of the fundamental results of the differential calculus relating an increment of a function f(x) and the values of its derivative. In analytic terms,

f(b) — f(a) = (b — a)f’(c)

where c is some number satisfying the inequality a < c < b. Formula (1) is valid if the function f(x) is continuous on the segment [a, b] and has a derivative at each point of the interval (a, b). In geometric terms (see Figure 1), formula (1) states that the tangent to the curve y = f(x) at a suitable point [c, f(c)] is parallel to the chord passing through the points [a, f(a)] and [b, f(b)]. The mean value theorem was discovered by J. Lagrange in 1797.

Figure 1

Among the different generalizations of the mean value theorem, note Bonnet’s mean value formula

and its particular case, Cauchy’s mean value formula

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In Section 3 we give a general mean-value theorem for the fuzzy integral, and see its relevance to optimization with inexact constraints.
In this section we give a mean-value theorem for the fuzzy integral.
The next result is a counterpart of the mean-value theorem in the classical measure theory.
does not change its positivity on [m,M], applying the integral mean-value theorem on (19) we get that there exists [xi], [member of] [m, M] such that
does not change its positivity on [m,M], applying the integral mean-value theorem on (43) we get that there exists [xi], [member of] [m, M] such that
In the following two sections of our paper we give some further generalizations of the Jensen-type inequalities on time scales allowing negative weights, and we also give the mean-value theorems of the Lagrange and Cauchy type for the functionals obtained by taking the difference of the left-hand side and right-hand side of these new inequalities.
We subsequently get with the help of the mean-value theorem of Cauchy that
We can write with the help of the mean-value theorem of Lagrange that
Using the mean-value theorem, (3.1) and the estimate on [U.sup.(0)], we conclude that
Using the mean-value theorem and the equation for [Z.sup.(n)], we have
Using the mean-value theorem, from (2.4) and (4.6), conclude that W (P, [tau]) satisfies
Also, equations (A5) and (A6), along with the Mean-Value Theorem, imply (A9) [Mathematical Expression Omitted] for some [q.sub.1] in ([q.sub.11] [sup.*], [q [bar].sub.1]), (A10) [Mathematical Expression Omitted] for some [q.sub.2] in ([q [bar].sub.22], [q.sub.2] [sup.*]).

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