Cauchy's mean theorem definition of Cauchy's mean theorem in the Free Online Encyclopedia.

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 existsx₀,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.

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.

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Mean Value Theorem (redirected from Cauchy's mean theorem)

Mean Value Theorem
(redirected from Cauchy's mean theorem)