I want to know if $$f(x,y)=x^4-3x^3y^2+2x^2y^2-3x^2y^3+y^4$$ has a local minimum or maximumin in $(0,0)$
What i tried is find the first non-vanishing derivative $D^4f(0,0)$ and see if $D^4f(0,0)(h,...h)$ is positiv-/negative-/indefinite. My problem is that I could not calculate $D^4f(0,0)$. I tried to follow this but then i get a $1\times16$ matrix for $D^4f$
Using polar coordinates, we have $f(x,y)=(x^2+y^2)^2-3x^2y^2(x+y)=r^4(1-3r\cos^2\theta \sin^2\theta (\cos \theta +\sin\theta)) $ so $f(x,y)\ge r^4(1-6r)$ for $r^2=x^2+y^2$. So $f$ can't be negative close to the origin, and $0$ is a local minimum.