Maximise x + y on the unit circle
Optimising x+y subject to x^2+y^2 = 1 gives 1.414214 at (0.707107, 0.707107), with multiplier λ = 0.707107. Full worked solution and an editable calculator.
Optimising f = x+y subject to x^2+y^2 = 1 gives f = 1.414214 at (0.707107, 0.707107), with λ = 0.707107. The maximum is √2, reached where the gradient of x + y points straight out of the circle.
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 = 1
The maximum is √2, reached where the gradient of x + y points straight out of the circle.
Critical points
| x | y | λ | f | Type |
|---|---|---|---|---|
| 0.707107 | 0.707107 | 0.707107 | 1.414214 | maximum |
| -0.707107 | -0.707107 | -0.707107 | -1.414214 | minimum |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Maximise x + y on the unit circle"? ▾
The optimum is f = 1.414214 at (0.707107, 0.707107), with Lagrange multiplier λ = 0.707107. The maximum is √2, reached where the gradient of x + y points straight out of the circle.
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 = 1. That is 3 equations in 3 unknowns, counting λ.
What does λ mean here? ▾
λ = 0.707107 means that raising the constraint value from 1 by one unit would change the optimal value of f by roughly 0.707107.
About this Calculator
Optimising x+y subject to x^2+y^2 = 1 gives 1.414214 at (0.707107, 0.707107), with multiplier λ = 0.707107. Full worked solution and an editable calculator.