Is the origin max or min for $f(x,y)=x^4+\frac{1}{4}y^4+4xy^3+4x^2y^2$?

Question

The gradient is null in (0,0) but the hessian matrix is null. The function hasn't simmetry.Can I use Taylor expansion?

Answer 1

The function is symmetric in x, and looks like $x^4$ on the $y = 0$ line, so the origin can't be a maximum: it's either a minimum or a saddle point. Along the $x = 0$ line, it looks like $\frac{1}{4}y^4$, so again, it looks like a minimum here.

And along any $y = ax$ line, the function looks like $x^4(1 + \frac{a^4}{4} + 4a^3 + 4a^2)$. For $a = -3$, that's $x^4(1 + \frac{81}{4} - 108 + 36) = \frac{-203}{4}x^4$, so along this line, it looks like a maximum.

Thus, it is neither a minimum nor a maximum.