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.
