Question 117
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x) + \sin(f(x)) \ge x$, for all $x \in \mathbb{R}$. Prove that
$$
\int_{0}^{\pi} f(x) \, dx \geq \frac{\pi^2}{2} - 2.
$$
$$
\int_{0}^{\pi} f(x) \, dx \geq \frac{\pi^2}{2} - 2.
$$