Extrema with Constraints

An extremum subject to a constraint is a largest or smallest value of a specified part of some process. In the language of mathematics, this amounts to asking for the extrema of a function on a specified part of its domain. - Below, we first give the precise definition. Then we go on to show how to use derivatives to detect extrema subject to a constraint.

We now present two methods which allow us, in certain circumstances, to detect extrema subject to a specified constraint: Firstly, the method of parametrizing the constraint set and, secondly, the method of Lagrange multipliers.

The method of parametrizing the constraint set This method is available whenever we can find a parametrization of the given constraint set. Here are the details:

Here is an example which shows how to use this theorem.

The method of Lagrange multipliers This method is available whenever we can find a function which has the constraint set as one of its level sets. Here are the details:

The number l is called a Lagrange multiplier. It helps us detect possible places where f can have an extremum subject to the given constraint. Here is an example which illustrates how this goes.

We now have two method to analyze a function for extrema on certain constraint sets. Each has its strengths and its weaknesses.

Sometimes we need to combine several methods to answer an extremum problem subject to a constraint. In this case the following theorem can be helpful, because it assures us of the existence of absolute extrema in certain situations.

Exercises