Question:

Let $f(x)$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Number of correct statements: I) If $f(x)$ strictly increasing on $(a,b)$ then $f'(x)\ge 0$ for all $x\in(a,b)$. II) If $f(x)$ strictly decreasing on $(a,b)$ then $f'(x)<0$ for all $x\in(a,b)$. III) $f(x)$ and $f'(x)$ have opposite sign for all $x$, then $f^2(x)$ is decreasing. IV) $f(x)$ and $f'(x)$ have opposite sign for all $x$, then $|f(x)|$ is increasing.