(Sorry if this question, along with some other recent questions, is trivial or there's errors in how I worded it. I'm a beginner calculus student)
Let $f:[a,b] \rightarrow \mathbb{R}$ be a continuous function twice-differentiable on the open interval $(a,b)$.
Proposition: if $c \in (a,b)$ is a strict local minimum of $f$, there exists a $\delta >0$ such that $f'$ is monotonically increasing in $(c, c + \delta)$.
Question 1: does this proposition hold?
Question 2: Let us consider a new function $g$ defined similarly to how $f$ was. Let $x_0$ satisfy $g'(x_0)=0$ and $g''(x_0)>0$. Moreover, there exists no $\delta >0$ such that $g'$ is monotonically increasing in $(x_0, x_0 + \delta)$. Then per contraposition on our original proposition, $x_0$ is not a local minimum of $g$. But this contradicts the 2nd derivative test. What's going on?
Note for $g$ to satisfy what we want, $g'$ could be some pathological differentiable function . Probably something you get after playing around with $\sin(\frac{1}{x})$; $g$ would of course be its primitive, surely not something expressible in elementary terms.