Question

Find the extreme values of

x^4+y^4-2(x-y)^2

Answer 1

The matrix $H(0,0)$ is negative semidefinite sine

$\lang \begin{bmatrix} -4 & 4 \\ 4 & -4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}, \begin{bmatrix} x \\ y \end{bmatrix} \rang =-4(x-y)^2\le0$

with equality if x=y

This is a necessary condition for a local maximum, but not sufficient. Therefore, the test is inconclusive.

Indeed, (0,0) is a saddle point. If we approach (0,0) along x=y then we have

$f(x,x)=2x^4>0$

However, if we approach along $x=-y$ , then

$f(x,-x)=2x^4-8x^2<0$

near (0,0).

extrema at $x=(-\sqrt2,\sqrt2)$ and $y=(\sqrt2,-\sqrt2)$