Exercise 13, § 4

Let $f$ be a numerical function differentiable in 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 $f(b) - f(a) = (b-a)f'(c)$; now prove that $f$ is strictly convex in $I$ or strictly concave in $I$ (now prove that $f'$ is strictly monotone in $I$).