Putnam 1991 B6

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^x b^{1-x} \leq a \frac{\sinh ux}{\sinh u} + b \frac{\sinh u(1-x)}{\sinh u}$$ for all $u$ with $0 < |u| \leq c$ and for all $x$, $0 < x < 1$. (Note: $\sinh u = (e^u - e^{-u})/2$.)