Find the maximum and minimum volumes of a rectangular box whose surface area equals 8000 square cm and whose edge length (sum of lengths of all edges) is 480 cm

Please see the attached file.

Solution
f(x, y, z) = xyz, volume
g(x, y, z) = 2xy + 2yz + 2xz - 8000
h(x, y, z) = 4x + 4 y + 4 z - 480

We have equations (using Lagrange multipliers):
f(x, y, z) = 0
h(x, y, z) = 0
grad_f = L * grad_g + M * grad_h

v = [x, y, z],
grad_f = grad (f, v)
grad_h = grad (h, v)

grad_f = [yz, xz, xy]
grad_g = [2y + 2z, 2x + 2z, 2x + 2y]
grad_h = [4, 4, 4]

System of equations:

yz - L (2y + 2z) - 4 M = 0
xz - L (2x + 2z) - 4 M = 0
xy - L (2x + 2y) - 4 M = 0
2 xy + 2 yz + 2 xz - 8000 = 0
4 x + 4 y + 4 z - 480 = 0

The solution of the system of equations gives the above solution.