14.7 Optimization
Global Extrema
The minimum and maximum values of a function
on a given domain are called global extreme values.
Bounded Domain
A domain
is bounded if it is contained in a disk of radius centered at the origin for some .
Closed Domain
A domain
is closed if it contains all boundary points.
Theorem
Let
be a continuous function on a closed, bounded domain in . Then
takes on both a global max and global min value in - The extreme values occur either at critical points in the interior of
or at points on the boundary of
Steps for determining the global extreme values of a function
- Check if
is bounded and closed and is continuous - Check the interior critical points
- Check the boundary for special points
- To find the global max/min of
on we compare the z-values of
- interior critical points
- special points on the boundary