This section is about optimizing a function, such that you find the absolute minimum and maximum on a certain region in .
Types of Regions
Extreme Value Theorem
- Closed Region: A region in
that includes its boundary
- Open Region: A region in
that excludes any of the boundary points
- Bounded Region: A region in
that is contained in a disk.
If a function
is continuous in a closed, bounded region
, then there are 2 points
is the absolute minimum and
is the absolute maximum of the function in the region
- Find all the critical points that are inside the region .
- Find all the critical points on the boundary of region .
- Find the function values for all the critical points
- Compare all the function values to see which is the smallest, and which is the largest. The smallest is the absolute minimum & and the largest is the absolute maximum.