Maximise 2x + 3y on x² + y² = 13
Optimising 2*x+3*y subject to x^2+y^2 = 13 gives 13 at (2, 3), with multiplier λ = 0.5. Full worked solution and an editable calculator.
Optimising f = 2*x+3*y subject to x^2+y^2 = 13 gives f = 13 at (2, 3), with λ = 0.5. A linear objective on a circle: the optimum sits where the constraint normal lines up with (2, 3).
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^2+y^2 = 13
A linear objective on a circle: the optimum sits where the constraint normal lines up with (2, 3).
Critical points
| x | y | λ | f | Type |
|---|---|---|---|---|
| 2 | 3 | 0.5 | 13 | maximum |
| -2 | -3 | -0.5 | -13 | minimum |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Maximise 2x + 3y on x² + y² = 13"? ▾
The optimum is f = 13 at (2, 3), with Lagrange multiplier λ = 0.5. A linear objective on a circle: the optimum sits where the constraint normal lines up with (2, 3).
What system of equations does this produce? ▾
Setting ∇f = λ∇g gives one equation per variable (x, y), plus the constraint x^2+y^2 = 13. That is 3 equations in 3 unknowns, counting λ.
What does λ mean here? ▾
λ = 0.5 means that raising the constraint value from 13 by one unit would change the optimal value of f by roughly 0.5.
About this Calculator
Optimising 2*x+3*y subject to x^2+y^2 = 13 gives 13 at (2, 3), with multiplier λ = 0.5. Full worked solution and an editable calculator.