Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 1 / 9D / ii / Solution

For the two-function assertion, defineThis function has the required continuity and differentiability and satisfies $H(a)=H(b)=0$. Rolle's theorem therefore gives $c$ withThis proves the Cauchy mean value theorem even when either endpoint increment vanishes; no division was used.