A differentiable function $f :\mathbb R \to\mathbb R$ satisfies:
- $f(2) = 5$
- $f(3) = 8$
- $f'(x) \le 4 \sin^2(\pi x)$ for all $x \in\mathbb R$.
How many values of $s \ge 2$ satisfy $f(s) = s^2$ ?
I could conclude that $x < 3$ by the 3rd condition, and tried to use the Mean Value Theorem but found it hard to go on. Can anyone give some hints?