Guse lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane. x + 9y + 8z = 27

This basically comes down to maximizing subject to and enforcing . We have the Lagrangian



with partial derivatives (set equal to 0 to find critical points)



Solving the first equation for gives . Substituting this into the next two equations, we have




Now





So the vertex of the cuboid in the given plane that maximizes the cuboids volume is , giving a volume of .