Exercise 12, § 4

Let $f$ be a continuous real function on an open interval $I \subset \mathbf{R}$; suppose that for every pair $(a, b)$ of points of $I$ such that $a < b$ the graph of $f$ lies either entirely above or entirely below the segment $M_aM_b$ on the interval $[a, b]$. Show that $f$ is convex on all of $I$ or concave on all $I$ (if in $]a, b[$ there is a point $c$ such that $M_c$ lies strictly above the segment $M_aM_b$ show that for every $x \in I$ such that $x > a$ the graph of $f$ lies above the segment $M_aM_x$ on the interval $[a, x]$).