Exercise 14, § 4

Let $f$ be a convex real function and strictly monotone on an open interval $I \subset \mathbf{R}$; let $g$ be the inverse function of $f$ (defined on the interval $f(I)$). Show that if $f$ is decreasing (resp. increasing) on $I$, then $g$ is convex (resp. concave) on $f(I)$.