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
Note that when we say an “open interval around ” we mean that we can find some
interval ,
not including the endpoints, such that . Or, in other words, c will be contained somewhere inside the interval and will not be
either of the endpoints.
Also, we will collectively 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 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 . There may be larger or smaller values of the
function at some other place, but relative to ,
or local to , 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 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.
For the function shown in this graph we have relative
maximums at and . Both of these point are relative maximums
since they are interior to the domain shown and are the largest point on the
graph in some interval around the point.
We also have a relative minimum at since this point is interior to the domain and
is the lowest point on the graph in an interval around it. The far right end point, ,
will not be a relative minimum since it is an end point.
The function will have an absolute maximum at and an absolute minimum at . These two points are the largest and smallest
that the function will ever be. We can
also notice that the absolute extrema for a function will occur at either the
endpoints of the domain or at relative extrema.
We will use this idea in later sections so it’s more important than it
might seem at the present time.
Let’s take a quick look at some examples to make sure that
we have the definitions of absolute extrema and relative extrema straight.
Example 1 Identify
the absolute extrema and relative extrema for the following function.
Solution
Since this function is easy enough to graph let’s do
that. However, we only want the graph
on the interval [1,2]. Here is the
graph,
Note that we used dots at the end of the graph to remind
us that the graph ends at these points.
We can now identify the extrema from the graph. It looks like we’ve got a relative and
absolute minimum of zero at and an absolute maximum of four at . Note that is not a relative maximum since it is at the
end point of the interval.
This function doesn’t have any relative maximums.

As we saw in the previous example functions do not have to
have relative extrema. It is completely
possible for a function to not have a relative maximum and/or a relative
minimum.
Example 2 Identify
the absolute extrema and relative extrema for the following function.
Solution
Here is the graph for this function.
In this case we still have a relative and absolute minimum
of zero at . We also still have an absolute maximum of
four. However, unlike the first
example this will occur at two points, and .
Again, the function doesn’t have any relative maximums.

As this example has shown there can only be a single
absolute maximum or absolute minimum value, but they can occur at more than one
place in the domain.
Example 3 Identify
the absolute extrema and relative extrema for the following function.
Solution
In this case we’ve given no domain and so the assumption
is that we will take the largest possible domain. For this function that means all the real
numbers. Here is the graph.
In this case the graph doesn’t stop increasing at either
end and so there are no maximums of any kind for this function. No matter which point we pick on the graph
there will be points both larger and smaller than it on either side so we can’t
have any maximums (of any kind, relative or absolute) in a graph.
We still have a relative and absolute minimum value of
zero at .

So, some graphs can have minimums but not maximums. Likewise, a graph could have maximums but not
minimums.
Example 4 Identify
the absolute extrema and relative extrema for the following function.
Solution
Here is the graph for this function.
This function has an absolute maximum of eight at and an absolute minimum of negative eight at
. This function has no relative extrema.

So, a function doesn’t have to have relative extrema as this
example has shown.
Example 5 Identify
the absolute extrema and relative extrema for the following function.
Solution
Again, we aren’t restricting the domain this time so
here’s the graph.
In this case the function has no relative extrema and no
absolute extrema.

As we’ve seen in the previous example functions don’t have
to have any kind of extrema, relative or absolute.
Example 6 Identify
the absolute extrema and relative extrema for the following function.
Solution
We’ve not restricted the domain for this function. Here is the graph.
Cosine has extrema (relative and absolute) that occur at
many points. Cosine has both relative
and absolute maximums of 1 at
Cosine also has both relative and absolute minimums of 1
at

As this example has shown a graph can in fact have extrema
occurring at a large number (infinite in this case) of points.
We’ve now worked quite a few examples and we can use these
examples to see a nice fact about absolute extrema. First let’s notice that all the functions
above were continuous functions. Next notice that every time we restricted the
domain to a closed interval (i.e. the
interval contains its end points) we got absolute maximums and absolute
minimums. Finally, in only one of the
three examples in which we did not restrict the domain did we get both an
absolute maximum and an absolute minimum.
These observations lead us the following theorem.
Extreme Value Theorem
So, if we have a continuous function on an interval [a,b] then we are guaranteed to have both
an absolute maximum and an absolute minimum for the function somewhere in the
interval. The theorem doesn’t tell us
where they will occur or if they will occur more than once, but at least it
tells us that they do exist somewhere.
Sometimes, all that we need to know is that they do exist.
This theorem doesn’t say anything about absolute extrema if
we aren’t working on an interval. We saw
examples of functions above that had both absolute extrema, one absolute
extrema, and no absolute extrema when we didn’t restrict ourselves down to an
interval.
The requirement that a function be continuous is also
required in order for us to use the theorem.
Consider the case of
Here’s the graph.
This function is not continuous at as we move in towards zero the function is approaching infinity. So, the function
does not have an absolute maximum. Note
that it does have an absolute minimum however.
In fact the absolute minimum occurs twice at both and .
If we changed the interval a little to say,
the function would now have both absolute extrema. We may only run into problems if the interval
contains the point of discontinuity. If
it doesn’t then the theorem will hold.
We should also point out that just because a function is not
continuous at a point that doesn’t mean that it won’t have both absolute
extrema in an interval that contains that point. Below is the graph of a function that is not
continuous at a point in the given interval and yet has both absolute extrema.
This graph is not continuous at ,
yet it does have both an absolute maximum ( ) and an absolute minimum ( ).
Also note that, in this case one of the absolute extrema occurred at the
point of discontinuity, but it doesn’t need to.
The absolute minimum could just have easily been at the other end point
or at some other point interior to the region.
The point here is that this graph is not continuous and yet does have
both absolute extrema
The point of all this is that we need to be careful to only
use the Extreme Value Theorem when the conditions of the theorem are met and
not misinterpret the results if the conditions aren’t met.
In order to use the Extreme Value Theorem we must have an
interval and the function must be continuous on that interval. If we don’t have an interval and/or the
function isn’t continuous on the interval then the function may or may not have
absolute extrema.
We need to discuss one final topic in this section before
moving on to the first major application of the derivative that we’re going to
be looking at in this chapter.
Fermat’s Theorem
Note that we can say that because we are also assuming that exists.
This theorem tells us that there is a nice relationship
between relative extrema and critical points.
In fact it will allow us to get a list of all possible relative
extrema. Since a relative extrema must
be a critical point the list of all critical points will give us a list of all
possible relative extrema.
Consider the case of . We saw that this function had a relative
minimum at in several earlier examples. So according to Fermat’s theorem should be a critical point. The derivative of the function is,
Sure enough is a critical point.
Be careful not to misuse this theorem. It doesn’t say that a critical point will be
a relative extrema. To see this,
consider the following case.
Clearly is a critical point. However we saw in an earlier example this
function has no relative extrema of any kind.
So, critical points do not have to be relative extrema.
Also note that this theorem says nothing about absolute
extrema. An absolute extrema may or may
not be a critical point.
To see the proof of this theorem see the Proofs From Derivative
Applications section of the Extras chapter.