Absolute maxima and minima of a function of two variables

 Objective: Find the absolute maxima and minima of a function of two variables.

Absolute maxima and minima

To find the absolute maxima and minima for a function of one variable on a closed interval, we nee need the find the critical points over that interval and evaluate the function at the endpoints of the interval. In a function of two variables, the interval is replaced by a closed boundary set. A set is bounded if all of its points are found inside of a ball or a disk. In order to find the absolute extrema, we start by calculating the critical points on the boundary set and the corresponding critical values of the function f.  Then we find the maximum and minimum value of the function at the boundary set. The highest function value is the absolute maximum and the smallest function value is the absolute minimum. Before calculating these values, we need to assure that there is a possibility that they exist. The following theorem gives this assurance.

Theorem

Since we now know that a continuous function function f(x,y) defined on a closed bounded set attains its extreme values, we need to know how to find them. the following theorem allows this.

Theorem

Problem solving strategy

Finding the maximum and minimum value at the boundary set can be challenging. If the boundary set is a rectangle or a set of straight lines it is possible to parameterize the line segments and determine the maxima on each of these segments. The same approach can be used for other shapes such as circles and ellipses.

Example

Use the problem-solving strategy to find the absolute extrema of a function to determine the absolute extrema of the following function:

 

Solution

Practice