Are you allowed to assume that the largrest such parallelopiped has edges parallel to the axes? If so, then if a vertex in the first octant is (x,y,z) the eges have length 2x, 2y, and 2z. Maximize V= 8xyz subject to the constraint that .

Was there a reason for titling this "Green's theorem"? It doesn't seem to have anything to do with Green's theorem.