Largest box with surface area 24
Optimising x*y*z subject to 2*(x*y+y*z+x*z) = 24 gives 8 at (2, 2, 2), with multiplier λ = 0.5. Full worked solution and an editable calculator.
Optimising f = x*y*z subject to 2*(x*y+y*z+x*z) = 24 gives f = 8 at (2, 2, 2), with λ = 0.5. The answer is a cube with side 2 and volume 8 — symmetry shows up in the multiplier equations.
The quantity to maximise or minimise
Left-hand side of g = c
Right-hand side
The system to solve
∂/∂x: ∂f/∂x = λ · ∂g/∂x ∂/∂y: ∂f/∂y = λ · ∂g/∂y ∂/∂z: ∂f/∂z = λ · ∂g/∂z constraint: 2*(x*y+y*z+x*z) = 24
The answer is a cube with side 2 and volume 8 — symmetry shows up in the multiplier equations.
Critical points
| x | y | z | λ | f | Type |
|---|---|---|---|---|---|
| 2 | 2 | 2 | 0.5 | 8 | maximum |
| -2 | -2 | -2 | -0.5 | -8 | minimum |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Largest box with surface area 24"? ▾
The optimum is f = 8 at (2, 2, 2), with Lagrange multiplier λ = 0.5. The answer is a cube with side 2 and volume 8 — symmetry shows up in the multiplier equations.
What system of equations does this produce? ▾
Setting ∇f = λ∇g gives one equation per variable (x, y, z), plus the constraint 2*(x*y+y*z+x*z) = 24. That is 4 equations in 4 unknowns, counting λ.
What does λ mean here? ▾
λ = 0.5 means that raising the constraint value from 24 by one unit would change the optimal value of f by roughly 0.5.
About this Calculator
Optimising x*y*z subject to 2*(x*y+y*z+x*z) = 24 gives 8 at (2, 2, 2), with multiplier λ = 0.5. Full worked solution and an editable calculator.