1. Absolute Maximum and Minimum Values

2. Absolute Extreme Values

3. Definition

4. The Extreme-Value Theorem

5. Finding Absolute Extreme Values
The procedure we use for finding the absolute extreme values can be outlined as
follows:
Determine the critical points. These are the interior points at which the gradient is zero (the stationary points) and the interior points at which the gradient does not exist.
Determine the points on the boundary that can possibly give rise to extreme values. At this stage this is a one-variable process.
Evaluate f at the points found in Steps 1 and 2.
The greatest of the numbers found in Step 3 is the absolute maximum; the least is the absolute minimum.

6. Example

7. Example