Use 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 + 7y + 8z = 21

Define the Lagrangian (L) to be

... L = x*y*z + λ * (x + 7y + 8z - 21)

Then the partial derivatives are

... ∂L/∂x = yz + λ

... ∂L/∂y = xz + 7λ

... ∂L/∂z = xy + 8λ

... ∂L/∂λ = x + 7y + 8z - 21

Setting these to zero gives rise to the equations

... λ = - yz

... xz - 7yz = 0 ⇒ x = 7y

... xy - 8yz = 0 ⇒ x = 8z

Then

... x + x + x - 21 = 0

... x = 7

... y = 1

... z = 7/8

The maximum volume, xyz, is then 49/8 = 6 1/8.