How to check extrema if second derivative test fails I have to find minima and maxima of $f(x,y)=x^4+6y^2-4xy^3-1$
I found three points that could be extrema - $(0,0)$, $(1,1)$ and $(-1,-1)$
I already checked $(1,1)$ and $(-1,-1)$ with second derivative test, but determinant of Hesian at $(0,0)$ equals $0$ and I can't show that there is no extremum (but maybe there is?). What do I do now?

$x^4+6y^2-4xy^3=x^4+2y^2(3-2xy)$ is positive for $0<|xy|<1.5$, so $f(x,y)$ has the minimum $-1$ for $x=y=0$
