Method of Lagrange Multipliers
Method of Lagrange Multipliers
The extrema of a function under a constraint can be found using the method of Lagrange multipliers. A condition for an extremum can be expressed by , which means that the level curve gradient and the constraint gradient are parallel. The scalar is called a Lagrange multiplier.
f(x,y)
g(x,y)=k
∇f=λ∇g
∇f
∇g
λ
This Demonstration illustrates this method for over 150 different combinations of functions and constraints. It shows the relationship between the normalized level curve gradient and the normalized constraint gradient . The image on the left shows the level curves of the function and the constraint , while the image on the right shows the 3D view of the surface , with the constraint curve projected onto the surface. Use the slider to move the point around the constraint curve to observe the relation between the level curve and the constraint gradients as the point reaches an extremum. The movable point turns from green to red when the two vectors are parallel, signifying that the point is near a local extremum.
∇f
∇g
f(x,y)
g(x,y)=k
z=f(x,y)