# Mean Value Theorem

(redirected from*Cauchy's mean theorem*)

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## mean value theorem

[′mēn ′val·yü ‚thir·əm]*(x)*is continuous on the closed interval [

*a,b*] and differentiable on the open interval (

*a,b*), then there exists

*x*

_{0},

*a*<>

*x*

_{0}<>

*b*, such that ƒ(

*b*) - ƒ(

*a*) = (

*b*-

*a*)ƒ′(

*x*

_{0}). 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.

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

and its particular case, Cauchy’s mean value formula