Question 157
Prove that a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing if and only if
$$
(c-b) \int_{a}^{b} f(x) \, \mathrm{d} x \leq (b-a) \int_{b}^{c} f(x) \, \mathrm{d} x
$$
for all real numbers $a \lt b \lt c$.
$$
(c-b) \int_{a}^{b} f(x) \, \mathrm{d} x \leq (b-a) \int_{b}^{c} f(x) \, \mathrm{d} x
$$
for all real numbers $a \lt b \lt c$.