I have a function such that $f'''$ is continuous and let's suppose there's some point $a \in \mathbb{R}$ such that $f'(a) = f''(a) = 0$. Does $f'''(a)>0$ tell me anything about whether $a$ is a relative maximum/minimum or neither? I've had a look at higher-order derivative tests, which tells me that $a$ should be a strictly increasing point of inflection, but I would like to formally prove this.
Edit: As Manzano has pointed out, the general version follows from the Taylor expansion at point $a$. I'm trying to show the specific version for $f'''$ without resorting to Taylor.