I think that we can solve this problem without Lagrange optimization tools:

We have a circle center (0,0) with radius of sqrt(8), our mission is to find point, (x,y), on that circle, with max{x*y}, try to show that such point have the following property -- x=y.

Therefor, if x=y, we'll get,

2x^2=8

x=2

y=2

and 3xy=12.