While learning about limits and derivatives, I came accross the following problem on one of Stewart's exercises book. I've been trying to wrap my head around it but I haven't got anywhere useful:
- $ \forall x f''(x) $ exists
- $ \exists c \in \mathbb{R} $ / $\forall x \neq c, f'(x) > 0 \wedge f'(c) = 0 $
- Then, $(c, f(c))$ is an inflection point.
From this I can gather that:
- $f$ is continuous
- $f$ is increasing $\forall x \neq c$.
My intuition tells me that $(3)$ is false since I might be able to come up with a function defined by parts that contradicts the statement, but I haven't found a way to prove this. Any pointers would be greatly appreciated.