I'm trying to prove the following claim:
$f:\left[p,q\right]\:\to\mathbb R,\ f$ is differentiable at $p$ and $p$ is a minimum of $f$. prove or disprove: $f'(p) \ge 0$.
When I draw a graph, it seemed obvious that the claim is true. the minimum is at the left edge of the function, so the function must go up afterwards. but as I know, the derivative at minimum / maximum equals to $0$. So why am I being asked about $f'(p) \ge 0$ ?
Tryed to prove using rolle theorem, but I don't know if the function is differentiable at $q$ (and maybe it isn't).
$$f'(p) = \lim_{x \to p \\ x > p} \frac{\overbrace{f(x) - f(p)}^{\ge 0}}{\underbrace{x-p}_{\ge 0}} \ge 0$$