Minimum and Maximum Values
Many of our applications in this chapter will revolve around minimum and maximum values of a function.  While we can all visualize the minimum and maximum values of a function we want to be a little more specific in our work here.  In particular we want to differentiate between two types of minimum or maximum values.  The following definition gives the types of minimums and/or maximums values that we’ll be looking at.
 
Definition
1.	We say that f(x) has an absolute (or global) maximum at x=c if 
2.	 for every x in the domain we are working on. 
3.	We say that f(x) has a relative (or local) maximum at x=c if  for every x in some open interval around x=c.
4.	We say that f(x) has an absolute (or global) minimum at x=c if  for every x in the domain we are working on. 
5.	We say that f(x) has a relative (or local) minimum at x=c if  for every x in some open interval around x=c.
 
Also, we call the minimum and maximum points of a function the extrema of the function.  So, relative extrema will refer to the relative minimums and maximums while absolute extrema refer to the absolute minimums and maximums.
 
Now, let’s talk a little bit about the subtle difference between the absolute and relative in the definition above.  
 
We will have an absolute maximum or minimum at x=c provided f(c) is the largest or smallest value that the function will ever take on the domain that we are working on.  Also, when we say the “domain we are working on” this simply means the range of x’s that we have chosen to work with for a given problem.  There may be other values of x that we can actually plug into the function but have excluded them for some reason.
 
A relative maximum or minimum is slightly different.  All that’s required for a point to be a relative maximum or minimum is for that point to be a maximum or minimum in some interval of x’s around x=c.  There may be larger or smaller values of the function at some other place, but relative to x=c, or local to x=c,  f(c) is larger or smaller than all the other function values that are near it.
 
Note as well that in order for a point to be a relative extrema we must be able to look at function values on both sides of x=c to see if it really is a maximum or minimum at that point.  This means that relative extrema do not occur at the end points of a domain.  They can only occur interior to the domain.
 
There is actually some debate on the preceding point.  Some folks do feel that relative extrema can occur on the end points of a domain.  However, in this class we will be using the definition that says that they can’t occur at the end points of a domain.
 
It’s usually easier to get a feel for the definitions by taking a quick look at a graph.