Waves Under Control

Let $f(x) = \mathrm{e}^x \sin(rx)$ with $r \in \mathbb{N}^*$.

1. If $r = 1$, find the intervals on which $f(x)$ is monotonic. 2. Prove that for every positive real number $M$ there exist real numbers $a, b > M$ such that $|f(x)| \leqslant 1$ for all $x \in [a, b]$.