As always, we must check all critical points in the domain. For a differentiable function, the critical points are the boundary points and the points inside at which . In our case,

so the only time ever vanishes is when (and thus ) at a boundary point, leaving us with only boundary points for the maximum. Our domain is an elliptical region in the first quadrant bounded by the and axes. When or equals , , leaving us with the boundary points at . By Lagrange's Method, we know that if is a maximum, then

Solving this vector equation for and will give you a point at which attains its maximum.