In the previous section we saw how to use the derivative to
determine the absolute minimum and maximum values of a function. However, there is a lot more information
about a graph that can be determined from the first derivative of a
function. We will start looking at that
information in this section. The main
idea we’ll be looking at in this section we will be identifying all the
relative extrema of a function.
Let’s start this section off by revisiting a familiar topic
from the previous chapter. Let’s suppose
that we have a function, . We know from our work in the previous chapter
that the first derivative, ,
is the rate of change of the function.
We used this idea to identify where a function was increasing,
decreasing or not changing.
Before reviewing this idea let’s first write down the
mathematical definition of increasing and decreasing. We all know what the graph of an
increasing/decreasing function looks like but sometimes it is nice to have a
mathematical definition as well. Here it
is.
Definition
This definition will actually be used in the proof of the
next fact in this section.
Now, recall that in the previous chapter we constantly used
the idea that if the derivative of a function was positive at a point then the
function was increasing at that point and if the derivative was negative at a
point then the function was decreasing at that point. We also used the fact that if the derivative
of a function was zero at a point then the function was not changing at that
point. We used these ideas to identify
the intervals in which a function is increasing and decreasing.
The following fact summarizes up what we were doing in the
previous chapter.
Fact
The proof of this fact is in the Proofs From
Derivative Applications section of the Extras chapter.
Let’s take a look at an example. This example has two purposes. First, it will remind us of the
increasing/decreasing type of problems that we were doing in the previous
chapter. Secondly, and maybe more
importantly, it will now incorporate critical points into the solution. We didn’t know about critical points in the
previous chapter, but if you go back and look at those examples, the first step
in almost every increasing/decreasing problem is to find the critical points of
the function.
Example 1 Determine
all intervals where the following function is increasing or decreasing.
Solution
To determine if the function is increasing or decreasing
we will need the derivative.
Note that when we factored the derivative we first
factored a “1” out to make the rest of the factoring a little easier.
From the factored form of the derivative we see that we
have three critical points : ,
,
and . We’ll need these in a bit.
We now need to determine where the derivative is positive
and where it’s negative. We’ve done
this several times now in both the Review chapter and the previous
chapter. Since the derivative is a
polynomial it is continuous and so we know that the only way for it to change
signs is to first go through zero.
In other words, the only place that the derivative may change signs is at the critical
points of the function. We’ve now got
another use for critical points. So,
we’ll build a number line, graph the critical points and pick test points
from each region to see if the derivative is positive or negative in each
region.
Here is the number line and the test points for the
derivative.
Make sure that you test your points in the
derivative. One of the more common
mistakes here is to test the points in the function instead! Recall that we know that the derivative
will be the same sign in each region.
The only place that the derivative can change signs is at the critical
points and we’ve marked the only critical points on the number line.
So, it looks we’ve got the following intervals of increase
and decrease.

In this example we used the fact that the only place that a
derivative can change sign is at the critical points. Also, the critical points for this function
were those for which the derivative was zero.
However, the same thing can be said for critical points where the
derivative doesn’t exist. This is nice
to know. A function can change signs
where it is zero or doesn’t exist. In
the previous chapter all our examples of this type had only critical points
where the derivative was zero. Now, that
we know more about critical points we’ll also see an example or two later on
with critical points where the derivative doesn’t exist.
Now that we have the previous “reminder” example out of the
way let’s move into some new material.
Once we have the intervals of increasing and decreasing for a function
we can use this information to get a sketch of the graph. Note that the sketch, at this point, may not
be super accurate when it comes to the curvature of the graph, but it will at
least have the basic shape correct. To
get the curvature of the graph correct we’ll need the information from the next
section.
Let’s attempt to get a sketch of the graph of the function
we used in the previous example.
Example 2 Sketch
the graph of the following function.
Solution
There really isn’t a whole lot to this example. Whenever we sketch a graph it’s nice to
have a few points on the graph to give us a starting place. So we’ll start by the function at the
critical points. These will give us
some starting points when we go to sketch the graph. These points are,
Once these points are graphed we go to the increasing and
decreasing information and start sketching.
For reference purposes here is the increasing/decreasing information.
Note that we are only after a sketch of the graph. As noted before we started this example we
won’t be able to accurately predict the curvature of the graph at this
point. However, even without this
information we will still be able to get a basic idea of what the graph
should look like.
To get this sketch we start at the very left of the graph
and knowing that the graph must be decreasing and will continue to decrease
until we get to . At this point the function will continue to
increase until it gets to . However, note that during the increasing
phase it does need to go through the point at and at this point we also know that the
derivative is zero here and so the graph goes through horizontally. Finally, once we hit the graph starts, and continues, to
decrease. Also, note that just like
at the graph will need to be horizontal when it
goes through the other two critical points as well.
Here is the graph of the function. We, of course, used a graphical program to
generate this graph, however, outside of some potential curvature issues if
you followed the increasing/decreasing information and had all the critical
points plotted first you should have something similar to this.

Let’s use the sketch from this example to give us a very
nice test for classifying critical points as relative maximums, relative
minimums or neither minimums or maximums.
Recall Fermat’s Theorem
from the Minimum and Maximum Values section.
This theorem told us that all relative extrema (provided the derivative
exists at that point of course) of a function will be critical points. The graph in the previous example has two
relative extrema and both occur at critical points as the Fermat’s Theorem
predicted. Note as well that we’ve got a
critical point that isn’t a relative extrema ( ).
This is okay since Fermat’s theorem doesn’t say that all critical points
will be relative extrema. It only states
that relative extrema will be critical points.
In the sketch of the graph from the previous example we can
see that to the left of the graph is decreasing and to the right of the graph is increasing and is a relative minimum. In other words, the graph is behaving around
the minimum exactly as it would have to be in order for to be a minimum. The same thing can be said for the relative
maximum at .
The graph is increasing on the left and decreasing on the right exactly
as it must be in order for to be a maximum. Finally, the graph is increasing on both
sides of and so this critical point can’t be a minimum
or a maximum.
These ideas can be generalized to arrive at a nice way to
test if a critical point is a relative minimum, relative maximum or
neither. If is a critical point and the function is
decreasing to the left of and is increasing to the right then must be a relative minimum of the
function. Likewise, if the function is
increasing to the left of and decreasing to the right then must be a relative maximum of the
function. Finally, if the function is
increasing on both sides of or decreasing on both sides of then can be neither a relative minimum nor a
relative maximum.
These ideas can be summarized up in the following test.
First Derivative Test
It is important to note here that the first derivative test
will only classify critical points as relative extrema and not as absolute
extrema. As we recall from the Finding Absolute Extrema section absolute extrema are
largest and smallest function values and may not even exist or be critical
points if they do exist.
The first derivative test is exactly that, a test using the
first derivative. It doesn’t ever use
the value of the function and so no conclusions can be drawn from the test
about the relative “size” of the function at the critical points (which would
be needed to identify absolute extrema) and can’t even begin to address the
fact that absolute extrema may not occur at critical points.
Let’s take at another example.
Example 3 Find
and classify all the critical points of the following function. Give the intervals where the function is
increasing and decreasing.
Solution
First we’ll need the derivative so we can get our hands on
the critical points. Note as well that
we’ll do some simplification on the derivative to help us find the critical
points.
So, it looks like we’ll have four critical points
here. They are,
Finding the intervals of increasing and decreasing will
also give the classification of the critical points so let’s get those
first. Here is a number line with the
critical points graphed and test points.
So, it looks like we’ve got the following intervals of
increasing and decreasing.
From this it looks like and are neither relative minimum or relative
maximums since the function is increasing on both side of them. On the other hand, is a relative maximum and is a relative minimum.
For completeness sake here is the graph of the function.

In the previous example the two critical points where the
derivative didn’t exist ended up not being relative extrema. Do not read anything into this. They often will be relative extrema. Check out this example in the Absolute Extrema
to see an example of one such critical point.
Let’s work a couple more examples.
Example 4 Suppose
that the elevation above sea level of a road is given by the following
function.
where x is in
miles. Assume that if x is positive we are to the east of
the initial point of measurement and if x
is negative we are to the west of the initial point of measurement.
If we start 25 miles to the west of the initial point of
measurement and drive until we are 25 miles east of the initial point how
many miles of our drive were we driving up an incline?
Solution
Okay, this is just a really fancy way of asking what the
intervals of increasing and decreasing are for the function on the interval
[25,25]. So, we first need the
derivative of the function.
Setting this equal to zero gives,
The solutions to this and hence the critical points are,
I’ll leave it to you to check that the critical points
that fall in the interval that we’re after are,
Here is the number line with the critical points and test
points.
So, it looks like the intervals of increasing and
decreasing are,
Notice that we had to end our intervals at 25 and 25
since we’ve done no work outside of these points and so we can’t really say
anything about the function outside of the interval [25,25].
From the intervals we can actually answer the question.
We were driving on an incline during the intervals of increasing and
so the total number of miles is,
Even though the problem didn’t ask for it we can also
classify the critical points that are in the interval [25,25].

Example 5 The
population of rabbits (in hundreds) after t
years in a certain area is given by the following function,
Determine if the population ever decreases in the first
two years.
Solution
So, again we are really after the intervals and increasing
and decreasing in the interval [0,2].
We found the only critical point to this function back in
the Critical Points section
to be,
Here is a number line for the intervals of increasing and
decreasing.
So, it looks like the population will decrease for a short
period and then continue to increase forever.
Also, while the problem didn’t ask for it we can see
that the single critical point is a
relative minimum.

In this section we’ve seen how we can use the first
derivative of a function to give us some information about the shape of a graph
and how we can use this information in some applications.
Using the first derivative to give us information about a
whether a function is increasing or decreasing is a very important application
of derivatives and arises on a fairly regular basis in many areas.