Lagrange Multipliers

Because we know both and we can calculate both of the gradient functions and using the level curve of as a second equation we can solve for the Lagrange Multipliers , , and .

Note that while there appears to be only two equations above, there are actually three due to the gradient functions being vectors:

Then, using the and we found using the system of equations we can then evaluate our original function to figure out which value is a minimum and maximum. The smaller value will be the minimum, the larger the maximum.

See Multivariable Calculus Notes - Chapter 14 - Sections 7-8 for source, examples, and more information.

• Points where a function has a minimum or maximum are called critical points.
• Like the Method of Lagrange Multipliers, extreme value theorem is a method to find absolute minimum and maximum points within a region.