Find an example of an even function such that $f'(0) = 0$ and $f(0)$ is not a local maximum / minimum of the function
I have been thinking about it for a while and I do not think that such function does exist. If we want the function not to have an extremum at $x = 0$, then the only possibility we are left with is that $(0,f(0))$ is an inflection point of the function. It means that the $n$th derivative ($n$ - odd) does not equal zero and all of its previous derivatives do. But then, since $n$ is odd, then the function, too, is odd - and so this does not work.
Can you think of an example of such function?
$$ f(x) = x^2 \sin \left( \frac{1}{x^2} \right) $$ when $x \neq 0,$ also $f(0) = 0$