second mean-value theorem

second mean-value theorem

[¦sek·ənd ¦mēn ¦val·yü ‚thir·əm]
(mathematics)
The theorem that for two functions ƒ(x) and g (x) that are continuous on a closed interval [a, b ] and differentiable on the open interval (a, b), such that g (b) ≠ g (a), there exists a number x1 in (a, b) such that either [ƒ(b) - ƒ(a)]/[g (b) -g (a)] = ƒ′(x1)/ g ′(x1) or ƒ′(x1) = g ′(x1) = 0. Also known as Cauchy's mean-value theorem; double law of the mean; extended mean-value theorem; generalized mean-value theorem.