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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 ?

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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.

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Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).

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Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.

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  • $\begingroup$ You are welcome. $\endgroup$ – MASL Sep 18 '15 at 16:55

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