[Math] Increasing Function $\implies$ Positive Derivative

Question

If $f$ is increasing in $[a,b]$ and is differentiable in $(a, b)$, then $f'(x)>0$ in $(a, b)$.

My thoughts: Even if a function is increasing, it can be increasing on the $2$nd and $3$rd quadrant, so the derivative can be negative.

Answer 1

Can't be negative, but it might be $0$; viz. $f(x)=x^3$ on $[-1,1]$, with derivative equal to $0$ at $x=0$.