Maximise xy subject to x + y = 10
Optimising x*y subject to x+y = 10 gives 25 at (5, 5), with multiplier λ = 5. Full worked solution and an editable calculator.
Optimising f = x*y subject to x+y = 10 gives f = 25 at (5, 5), with λ = 5. The classic fence problem: a fixed perimeter encloses the most area when the sides 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 constraint: x+y = 10
The classic fence problem: a fixed perimeter encloses the most area when the sides are equal.
Critical points
| x | y | λ | f | Type |
|---|---|---|---|---|
| 5 | 5 | 5 | 25 | stationary point |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Maximise xy subject to x + y = 10"? ▾
The optimum is f = 25 at (5, 5), with Lagrange multiplier λ = 5. The classic fence problem: a fixed perimeter encloses the most area when the sides are equal.
What system of equations does this produce? ▾
Setting ∇f = λ∇g gives one equation per variable (x, y), plus the constraint x+y = 10. That is 3 equations in 3 unknowns, counting λ.
What does λ mean here? ▾
λ = 5 means that raising the constraint value from 10 by one unit would change the optimal value of f by roughly 5.
About this Calculator
Optimising x*y subject to x+y = 10 gives 25 at (5, 5), with multiplier λ = 5. Full worked solution and an editable calculator.