I need a little help with finding absolute extrema bounded by a region.

Find the absolute extrema of $\displaystyle G(x,y) = xy$ over the region $\displaystyle 4x^2+y^2\leq2$.

$\displaystyle G_{x}= y$

$\displaystyle G_{y}= x$

Therefore:

$\displaystyle y=0$ and $\displaystyle x=0$

$\displaystyle G(0,0) = 0$

Where do I go from here and how?