Visualizing Lagrange Multipliers

Method of Lagrange Multipliers

Suppose

and

are real-valued, differentiable functions with continuous partial derivatives, c is a real number, and

does not vanish on the set

. Then the extreme values of

subject to the constraint

can be found among the solutions to the system of equations

Geometrically, this means that extreme values of

on the set

occur 1) where

vanishes, or 2) where

and

are parallel.

This applet gives a visual explanation for why the method of Lagrange multipliers correctly identifies extrema of a function

restricted to the constraint

. First enter a function

Displayed above is a contour plot of several level sets of

, colored in rainbow order by height. Also displayed is a level set

, which you can change by moving the value of k on the appropriate slider. Next, you can enter a second function

in the appropriate box and a value

on the appropriate slider. The constraint set

will display above, as well as a point

on an intersection of your constraint set

and your chosen level set

. Finally, toggling checkboxes shows or hides the gradients of f and g at the point

. The largest value of

constrained to

by definition should be on the highest level set of

that intersects the curve

. By moving the level set

, you should see that this intersection occurs either 1) where

vanishes, or 2) where

and

are parallel.