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.

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