 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ia/paper-1/9d/ii/solution

For the two-function assertion, define

H (x) = [f (b) - f (a)] [g (x) - g (a)] - [g (b) - g (a)] [f (x) - f (a)] .

This function has the required continuity and differentiability and satisfies

H (a) = H (b) = 0

. Rolle's theorem therefore gives

c

with

[f (b) - f (a)] g^{'} (c) = [g (b) - g (a)] f^{'} (c) .

This proves the Cauchy mean value theorem even when either endpoint increment vanishes; no division was used.

 New to topics? Read the docs here!