So I have to find the absolute extrema of f(x,y)=xy on D={(x,y)|x^2+2y^2<=4}

I took partial derivatives of xy to find a critical point (0,0). Now I have to find the critical points on the boundary. This is where x^2+2y^2=4. I tried following this (Extrema for Multivariable Functions - Classwiki) so I rearranged it into x=sqrt(4-2y^2) and subbed it into f(x,y).

I now have f(x,y) = y*sqrt(4-2y^2). I'm not sure where to go from here. The example above is a bit unclear but I think I make that = 0. This means y = 0 or +-sqrt(2). But if you put those into f(x,y), they all come out as 0. So I have no idea if what I'm doing is right or if the method is even right. Any help would be appreciated.