In the previous section we saw how we could use the first
derivative of a function to get some information about the graph of a
function. In this section we are going
to look at the information that the second derivative of a function can give us
a about the graph of a function.
Before we do this we will need a couple of definitions out
of the way. The main concept that we’ll
be discussing in this section is concavity.
Concavity is easiest to see with a graph (we’ll give the mathematical
definition in a bit).
So a function is concave
up if it “opens” up and the function is concave down if it “opens” down.
Notice as well that concavity has nothing to do with increasing or
decreasing. A function can be concave up
and either increasing or decreasing.
Similarly, a function can be concave down and either increasing or
decreasing.
It’s probably not the best way to define concavity by saying
which way it “opens” since this is a somewhat nebulous definition. Here is the mathematical definition of
concavity.
Definition 1
To show that the graphs above do in fact have concavity
claimed above here is the graph again (blown up a little to make things
clearer).
So, as you can see, in the two upper graphs all of the
tangent lines sketched in are all below the graph of the function and these are
concave up. In the lower two graphs all
the tangent lines are above the graph of the function and these are concave
down.
Again, notice that concavity and the increasing/decreasing
aspect of the function is completely separate and do not have anything to do
with the other. This is important to
note because students often mix these two up and use information about one to
get information about the other.
There’s one more definition that we need to get out of the
way.
Definition 2
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A point  is called an inflection point if the function is continuous at the point and
the concavity of the graph changes at that point.
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Now that we have all the concavity definitions out of the
way we need to bring the second derivative into the mix. We did after all start off this section
saying we were going to be using the second derivative to get information about
the graph. The following fact relates
the second derivative of a function to its concavity. The proof of this fact is in the Proofs From
Derivative Applications section of the Extras chapter.
Fact
Notice that this fact tells us that a list of possible
inflection points will be those points where the second derivative is zero or
doesn’t exist. Be careful however to not
make the assumption that just because the second derivative is zero or doesn’t
exist that the point will be an inflection point. We will only know that it is an inflection
point once we determine the concavity on both sides of it. It will only be an inflection point if the
concavity is different on both sides of the point.
Now that we know about concavity we can use this information
as well as the increasing/decreasing information from the previous section to
get a pretty good idea of what a graph should look like. Let’s take a look at an example of that.
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Example 1 For
the following function identify the intervals where the function is
increasing and decreasing and the intervals where the function is concave up
and concave down. Use this information
to sketch the graph.
Solution
Okay, we are going to need the first two derivatives so
let’s get those first.
Let’s start with the increasing/decreasing information
since we should be fairly comfortable with that after the last section.
There are three critical points for this function : ,
,
and . Below is the number line for the
increasing/decreasing information.
So, it looks like we’ve got the following intervals of
increasing and decreasing.
Note that from the first derivative test we can also say
that is a relative maximum and that is a relative minimum. Also is neither a relative minimum or maximum.
Now let’s get the intervals where the function is concave
up and concave down. If you think
about it this process is almost identical to the process we use to identify
the intervals of increasing and decreasing.
This only difference is that we will be using the second derivative
instead of the first derivative.
The first thing that we need to do is identify the
possible inflection points. These will
be where the second derivative is zero or doesn’t exist. The second derivative in this case is a
polynomial and so will exist everywhere.
It will be zero at the following points.
As with the increasing and decreasing part we can draw a
number line up and use these points to divide the number line into
regions. In these regions we know that
the second derivative will always have the same sign since these three
points are the only places where the function may change sign.
Therefore, all that we need to do is pick a point from each region and
plug it into the second derivative.
The second derivative will then have that sign in the whole region
from which the point came from
Here is the number line for this second derivative.
So, it looks like we’ve got the following intervals of
concavity.
This also means that
are all inflection points.
All this information can be a little overwhelming when
going to sketch the graph. The first
thing that we should do is get some starting points. The critical points and inflection points
are good starting points. So, first
graph these points. Now, start to the
left and start graphing the increasing/decreasing information as we did in
the previous section when all we had was the increasing/decreasing information. As we graph this we will make sure that the
concavity information matches up with what we’re graphing.
Using all this information to sketch the graph gives the
following graph.
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We can use the previous example to illustrate another way
to classify some of the critical points of a function as relative maximums or
relative minimums.
Notice that is a relative maximum and that the function is
concave down at this point. This means that must be negative. Likewise, is a relative minimum and the function is
concave up at this point. This means
that must be positive.
As we’ll see in a bit we will need to be very careful with . In this case the second derivative is zero,
but that will not actually mean that is not a relative minimum or maximum. We’ll see some examples of this in a bit, but
we need to get some other information taken care of first.
It is also important to note here that all of the critical
points in this example were critical points in which the first derivative was
zero and this is required for this to work.
We will not be able to use this test on critical points where the
derivative doesn’t exist.
Here is the test that can be used to classify some of the
critical points of a function. The proof
of this test is in the Proofs
From Derivative Applications section of the Extras chapter.
Second Derivative
Test
The third part of the second derivative test is important to
notice. If the second derivative is zero
then the critical point can be anything.
Below are the graphs of three functions all of which have a critical
point at ,
the second derivative of all of the functions is zero at and yet all three possibilities are exhibited.
The first is the graph of . This graph has a relative minimum at .
Next is the graph of which has a relative maximum at .
Finally, there is the graph of and this graph had neither a relative minimum
or a relative maximum at .
So, we can see that we have to be careful if we fall into
the third case. For those times when we
do fall into this case we will have to resort to other methods of classifying
the critical point. This is usually done
with the first derivative test.
Let’s go back and relook at the critical points from the
first example and use the Second Derivative Test on them, if possible.
Let’s work one more example.
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Example 3 For
the following function find the inflection points and use the second
derivative test, if possible, to classify the critical points. Also, determine the intervals of
increase/decrease and the intervals of concave up/concave down and sketch the
graph of the function.
Solution
We’ll need the first and second derivatives to get us
started.
The critical points are,
Notice as well that we won’t be able to use the second
derivative test on to classify this critical point since the
derivative doesn’t exist at this point.
To classify this we’ll need the increasing/decreasing information that
we’ll get to sketch the graph.
We can however, use the Second Derivative Test to classify
the other critical point so let’s do that before we proceed with the
sketching work. Here is the value of
the second derivative at .
So, according to the second derivative test is a relative maximum.
Now let’s proceed with the work to get the sketch of the
graph and notice that once we have the increasing/decreasing information
we’ll be able to classify .
Here is the number line for the first derivative.
So, according to the first derivative test we can verify
that is in fact a relative maximum. We can also see that is a relative minimum.
Be careful not to assume that a critical point that can’t
be used in the second derivative test won’t be a relative extrema. We’ve clearly seen now both with this
example and in the discussion after we have the test that just because we
can’t use the Second Derivative Test or the Test doesn’t tell us anything
about a critical point doesn’t mean that the critical point will not be a
relative extrema. This is a common
mistake that many students make so be careful when using the Second
Derivative Test.
Okay, let’s finish the problem out. We will need the list of possible
inflection points. These are,
Here is the number line for the second derivative. Note that we will need this to see if the
two points above are in fact inflection points.
So, the concavity only changes at and so this is the only inflection point for
this function.
Here is the sketch of the graph.
The change of concavity at is hard to see, but it is there it’s just a
very subtle change in concavity.
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