Minimise x² + y² + z² on the plane x + y + z = 3
Optimising x^2+y^2+z^2 subject to x+y+z = 3 gives 3 at (1, 1, 1), with multiplier λ = 2. Full worked solution and an editable calculator.
Optimising f = x^2+y^2+z^2 subject to x+y+z = 3 gives f = 3 at (1, 1, 1), with λ = 2. The closest point on a plane to the origin lies along the plane's normal direction.
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 = 3
The closest point on a plane to the origin lies along the plane's normal direction.
Critical points
| x | y | z | λ | f | Type |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 2 | 3 | stationary point |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Minimise x² + y² + z² on the plane x + y + z = 3"? ▾
The optimum is f = 3 at (1, 1, 1), with Lagrange multiplier λ = 2. The closest point on a plane to the origin lies along the plane's normal direction.
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 = 3. That is 4 equations in 4 unknowns, counting λ.
What does λ mean here? ▾
λ = 2 means that raising the constraint value from 3 by one unit would change the optimal value of f by roughly 2.
About this Calculator
Optimising x^2+y^2+z^2 subject to x+y+z = 3 gives 3 at (1, 1, 1), with multiplier λ = 2. Full worked solution and an editable calculator.