The Geometry of Lagrange Multipliers

show

3D

2D

point

f(x,y) = 0.866025

near max

Move a point around a constraint curve to see the relationship between the blue gradient of the function

f(x,y)

to be optimized and the red gradient of the function

g(x,y)

for the constraint

g(x,y)=0

. The orange vector is the projection of the gradient of

f

onto the tangent line of the constraint curve; its direction is the direction of increase along the constraint and its magnitude is the slope (that is, the directional derivative) of

f

in that direction. There are both 2D and 3D views, with the constraint curve laid out upon the graph of the function.

At a local extremum, the gradients of the functions

f

and

g

are parallel. Thus we get the condition for an extremum

∇f=λ∇g

, where

λ

is called a Lagrange multiplier.