# What are examples of $f''(\operatorname{critical})=0$ with local extrema?

The second derivative test of critical points shows the type of extreme at the critical point:

$$f''(\operatorname{critical})>0$$, then it's local minimum.

$$f''(\operatorname{critical})<0$$, then it's local maximum.

$$f''(\operatorname{critical})=0$$, it may or may not be local extreme.

I searched the web for examples of functions that have $$f''(\text{critical})=0$$ and it's local extreme, but didn't find any.

What are examples of $$f''(\operatorname{critical})=0$$ with local extrema ?

The usual simple answer is $f(x)=x^4$, which is obviously non-negative, so $x=0$ is a minimum. You can use similar ideas to cook up other more complicated examples for more specific situations.
Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).
Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.