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

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

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.