Can we get Cauchy mean value theorem from Lagrange mean value theorem?

Answer 1

The answer is yes. Cauchy's mean value theorem (CMVT) is a generalization of the Lagrange mean value theorem (LMVT), so we can prove the CMVT using the LVMT. Let f,g be continuous functions on the closed interval [a,b] which are also differentiable on (a,b). We wish to find a value c in (a,b) such that [f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c). First, suppose that g(b) = g(a). Then by the LMVT, there exists c in (a,b) such that g'(c) = (g(b) - g(a))/(b-a) = 0.This value of c satisfies the criteria for the CMVT in this case. Now suppose that g(b) =/= g(a). Define the function h on [a,b] via h(x) = f(x) - f(a) - (g(x)-g(a))*(f(b) - f(a))/(g(b)-g(a)). Now h is continuous on [a,b] and differentiable on (a,b) via the algebraic properties of continuous and differentiable functions. Observe that h(a) = h(b) = 0. Thus by the LMVT, there exists c in (a,b) such that (h(b) - h(a))/(b-a) = 0 = h'(c) = f'(c) - g'(c)*(f(b)-f(a))/(g(b)-g(a)). Rearranging this expression yields the result.