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.