Classify the critical point $(0,0)$ of the multivariable function $x^4+y^4-2x^2+4xy-2y^2$

Question

I have the following multivariate function before me:

$f(x,y)=x^4+y^4-2x^2+4xy-2y^2$

I have to find the relative maximum and minimum values of $f(x,y)$.

On solving first order conditions $f_x=f_y=0$, we get three critical points $(0,0),(-\sqrt{2},\sqrt{2}),(\sqrt{2},-\sqrt{2}).$

For points $(-\sqrt{2},\sqrt{2}),(\sqrt{2},-\sqrt{2})$,
we get:

$D=f_{xx}.f_{yy}-(f_{xy})^2=384>0$
and these points turn out to be relative minima.

But when it comes to the point$(0,0)$, we get $D=0$.

How can I tell from here whether point $(0,0)$ is a relative minima, maxima or a saddle point? Please help.

Answer 1

Here's a hint: Write $f(x,y) = -2 (x-y)^2 + x^4 + y^4$ and look at what the function is doing along the coordinate axes and along the line $y=x$.