Exercise 14, § 4

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