Maximise xyz subject to x + y + z = 12
Optimising x*y*z subject to x+y+z = 12 gives 64 at (4, 4, 4), with multiplier λ = 16. Full worked solution and an editable calculator.
Optimising f = x*y*z subject to x+y+z = 12 gives f = 64 at (4, 4, 4), with λ = 16. The AM-GM inequality in disguise: the product is largest when all three numbers are equal.
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: x+y+z = 12
The AM-GM inequality in disguise: the product is largest when all three numbers are equal.
Critical points
| x | y | z | λ | f | Type |
|---|---|---|---|---|---|
| 4 | 4 | 4 | 16 | 64 | maximum |
| 0 | 0 | 12 | 0 | 0 | minimum |
| 0 | 12 | 0 | 0 | 0 | minimum |
| 12 | 0 | 0 | 0 | 0 | minimum |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Maximise xyz subject to x + y + z = 12"? ▾
The optimum is f = 64 at (4, 4, 4), with Lagrange multiplier λ = 16. The AM-GM inequality in disguise: the product is largest when all three numbers are equal.
What system of equations does this produce? ▾
Setting ∇f = λ∇g gives one equation per variable (x, y, z), plus the constraint x+y+z = 12. That is 4 equations in 4 unknowns, counting λ.
What does λ mean here? ▾
λ = 16 means that raising the constraint value from 12 by one unit would change the optimal value of f by roughly 16.
About this Calculator
Optimising x*y*z subject to x+y+z = 12 gives 64 at (4, 4, 4), with multiplier λ = 16. Full worked solution and an editable calculator.