second mean-value theorem

second mean-value theorem [¦sek·ənd ¦mēn ¦val·yü ‚thir·əm]

(mathematics)

The theorem that for two functions ƒ(x) andg(x) that are continuous on a closed interval [a,b] and differentiable on the open interval (a,b), such thatg(b) ≠g(a), there exists a numberx₁in (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.