When does $g(x) > 0$ and $0 \le f'(x) \le g(x)f(x)$ imply that $f(x) = 0$?
This question is inspired by my urge to generalize this question:
Derivative bounded by the original function
I have some ideas on this, but I want to see the most general conditions on $g(x)$ (and, perhaps, $f(x)$) that can make $f(x)$ be zero.