Prove that $f$ is increasing if and only if a given inequality holds Let $f: [0, \infty) \to \mathbb{R}$ be a continuous function. Prove that $f$ is increasing if and only if:
$$\int_a^b f(x) dx \leq bf(b) - af(a), \, \forall \, \, 0 \leq a \leq b.$$
I have no difficulties in proving that if $f$ is increasing then the inequality holds. But I haven't figured out yet how to prove it the other way around, that is knowing the inequality and proving that $f$ is increasing.
Thank you!

Note that if $f(a) > f(b) $ then by continuity we can choose $c\in(a, b] $ such that $f(x) > f(c) $ for all $x\in [a, c) $ and hence $$\int_{a} ^{c} f(t) \, dt>(c-a) f(c) $$ And given condition on $f$ implies that integral above is not greater than $cf(c) - af(a) $. It now follows that $f(a) <f(c) $ which is contrary to $f(a) >f(c) $.  Thus we must have $f(a) \leq f(b) $. 
