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
