Section Summary: 15.8

Lagrange Multipliers

1. Definitions

2. Theorems

3. Properties/Tricks/Hints/Etc.

   Method of Lagrange Multipliers To find the extrema of f(x,y,z) sugject to the constraint g(x,y,z)=k (assuming that extrema exist),

   1. Find all values of x, y, z, and such that

      

      and g(x,y,z)=k.

   2. Evaluate f at all points (x,y,z) are solutions to step 1: the largest is the max, whereas the smallest is the min.

   An alternate formulation of Lagrange multipliers is that we seek extrema of a function

   

   Differentiate with respect to x, y, z, and ; what equations do you derive?

4. Summary

   Lagrange multipliers are simply means of introducing constraints into an optimization problem. Perhaps the easiest way to think of this is to imagine two surfaces intersecting, and to ask what the largest value of the first function is upon the curve of intersection.