Exercise 13, § 4

Let $f$ be a differentiable real function on an open interval $I \subset \mathbf{R}$. Suppose that for every pair $(a, b)$ of points of $I$ such that $a < b$ there exists a unique point $c \in ]a, b[$ such that $f(b) - f(a) = (b - a)f'(c)$; show that $f$ is strictly convex on $I$ or strictly concave on $I$ (show that $f'$ is strictly monotone on $I$).