Finding Maximums and Minimums
Read this section to learn about maximums, minimums, and extreme values for functions. Work through practice problems 1-5.
A Little Terminology
Before we examine how calculus can help us find maximums and minimums, we need to define the concepts we will develop and use.
Definitions: 
has a maximum or global maximum at a if 
for all 
in the domain of .
The maximum value
of is 
, and this maximum value of occurs at 
The maximum point on the graph of is 
. (Fig. 1)
Fig. 1
Definition: has a local or relative maximum at a if for all near 
or in some open interval which contains .
Global and local minimums are defined similarly by replacing the 
with 
in the previous definitions.
Definition: has a global extreme at if 
is a global maximum or minimum.
has a local extreme at if is a local maximum or minimum.
The local and global extremes of the function in Fig. 2 are labeled. You should notice that every global extreme is also a local extreme, but there are local extremes which are not global extremes. If 
is the height of the earth above sea level at the location , then the global maximum of 
is (summit of Mt. Everest) =29,028 feet. The local maximum of for the United States is (summit of Mt. McKinley)=20,320 feet. The local minimum of for the United States is (Death Valley)=-282 feet.
Practice 1: The table shows the annual calculus enrollments at a large university. Which years had relative maximum or minimum calculus enrollments? What were the global maximum and minimum enrollments in calculus?
