Lagrange multipliers
This figure demonstrates the method of Lagrange mulipliers to solve a problem like the following:
Suppose functions and are given. Find the local maximum and minimum values of subject to the constraint that .
Graphically, the equation defines a curve in the plane, call it ; we must find the point(s) on where reaches its extreme values.
When the figure loads, and are defined and you see curve , a point on , and the vector at the point. Keep in mind that points in the direction of greatest increase of . With this in mind, slowly drag the point around the curve - can you get a sense of where is increasing or decreasing along the curve? What makes attain a maximum or minimum value along the curve?
When reaches an extreme value on , the gradient must be perpendicular to - this is the Orthogonal Gradient Theorem. Since is always orthogonal to , this is equivalent to the equation for some scalar . By finding the points on where the equation is satisfied we can determine where takes on its extreme values.
Finally, in the figure you can draw level curves for and explore the relationship between level curves of , the curve , and the location of the extreme points.
Developed for use with Thomas' Calculus and Interactive Calculus, published by Pearson.