Question 11
Prove that a continuous function $f: \mathbb{R} \to \mathbb{R}$ is non-decreasing if and only if
$$
(c - b) \int_{a}^{b} f(x) \, dx \leq (b - a) \int_{b}^{c} f(x) \, dx,
$$
for all $a \lt b \lt c$.
$$
(c - b) \int_{a}^{b} f(x) \, dx \leq (b - a) \int_{b}^{c} f(x) \, dx,
$$
for all $a \lt b \lt c$.