Exercise 10, § 2

Let $f$ and $g$ be two finite real functions, strictly positive, continuous and differentiable on an open interval $I$. Show that if $f'$ and $g'$ are strictly positive and $f'/g'$ is strictly increasing on $I$, then either $f/g$ is strictly increasing on $I$, or else there exists a number c ∈ I such that $f/g$ is strictly decreasing for $x \leq c$ and strictly increasing for $x \geq c$ (note that if one has $f'(x)/g'(x) < f(x)/g(x)$ then also

for all $y < x$).