Solution
= 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
$$
\boxed{[f(b)-f(a)]g'(c)=[g(b)-g(a)]f'(c).}
$$
This proves the <Cauchy mean value theorem> even when either endpoint increment vanishes; no division was used.