Critical points will show up throughout a majority of this
chapter so we first need to define them and work a few examples before getting
into the sections that actually use them.
Definition
Note that we require that exists in order for to actually be a critical point. This is an important, and often overlooked,
point.
The main point of this section is to work some examples
finding critical points. So, let’s work
some examples.
Example 1 Determine
all the critical points for the function.
Solution
We first need the derivative of the function in order to
find the critical points and so let’s get that and notice that we’ll factor
it as much as possible to make our life easier when we go to find the
critical points.
Now, our derivative is a polynomial and so will exist
everywhere. Therefore the only
critical points will be those values of x
which make the derivative zero. So, we
must solve.
Because this is the factored form of the derivative it’s
pretty easy to identify the three critical points. They are,

Polynomials are usually fairly simple functions to find
critical points for provided the degree doesn’t get so large that we have
trouble finding the roots of the derivative.
Most of the more “interesting” functions for finding
critical points aren’t polynomials however.
So let’s take a look at some functions that require a little more effort
on our part.
Example 2 Determine
all the critical points for the function.
Solution
To find the derivative it’s probably easiest to do a
little simplification before we actually differentiate. Let’s multiply the root through the
parenthesis and simplify as much as possible.
This will allow us to avoid using the product rule when taking the
derivative.
Now differentiate.
We will need to be careful with this problem. When faced with a negative exponent it is
often best to eliminate the minus sign in the exponent as we did above. This isn’t really required but it can make
our life easier on occasion if we do that.
Notice as well that eliminating the negative exponent in
the second term allows us to correctly identify why is a critical point for this function. Once we move the second term to the
denominator we can clearly see that the derivative doesn’t exist at and so this will be a critical point. If you don’t get rid of the negative
exponent in the second term many people will incorrectly state that is a critical point because the derivative
is zero at . While this may seem like a silly point,
after all in each case is identified as a critical point, it is
sometimes important to know why a point is a critical point. In fact, in a couple of sections we’ll see
a fact that only works for critical points in which the derivative is zero.
So, we’ve found one critical point (where the derivative
doesn’t exist), but we now need to determine where the derivative is zero
(provided it is of course…). To help
with this it’s usually best to combine the two terms into a single rational
expression. So, getting a common
denominator and combining gives us,
Notice that we still have as a critical point. Doing this kind of combining should never
lose critical points, it’s only being done to help us find them. As we can see it’s now become much easier
to quickly determine where the derivative will be zero. Recall that a rational expression will only
be zero if its numerator is zero (and provided the denominator isn’t also
zero at that point of course).
So, in this case we can see that the numerator will be
zero if and so there are two critical points for
this function.

Example 3 Determine
all the critical points for the function.
Solution
We’ll leave it to you to verify that using the quotient
rule we get that the derivative is,
Notice that we factored a “1” out of the numerator to
help a little with finding the critical points. This negative out in front will not affect
the derivative whether or not the derivative is zero or not exist but will
make our work a little easier.
Now, we have two issues to deal with. First the derivative will not exist if
there is division by zero in the denominator.
So we need to solve,
We didn’t bother squaring this since if this is zero, then
zero squared is still zero and if it isn’t zero then squaring it won’t make
it zero.
So, we can see from this that the derivative will not
exist at and . However, these are NOT critical points
since the function will also not exist at these points. Recall that in order for a point to be a
critical point the function must actually exist at that point.
At this point we need to be careful. The numerator doesn’t factor, but that
doesn’t mean that there aren’t any critical points where the derivative is
zero. We can use the quadratic formula
on the numerator to determine if the fraction as a whole is ever zero.
So, we get two critical points. Also, these are not “nice” integers or
fractions. This will happen on
occasion. Don’t get too locked into
answers always being “nice”. Often
they aren’t.
Note as well that we only use real numbers for critical
points. So, if upon solving the
quadratic in the numerator, we had gotten complex number these would not have
been considered critical points.
Summarizing, we have two critical points. They are,
Again, remember that while the derivative doesn’t exist at
and neither does the function and so these two
points are not critical points for this function.

So far all the examples have not had any trig functions,
exponential functions, etc. in
them. We shouldn’t expect that to always
be the case. So, let’s take a look at
some examples that don’t just involve powers of x.
Example 4 Determine
all the critical points for the function.
Solution
First get the derivative and don’t forget to use the chain
rule on the second term.
Now, this will exist everywhere and so there won’t be any
critical points for which the derivative doesn’t exist. The only critical points will come from
points that make the derivative zero.
We will need to solve,
Solving this equation gives the following.
Don’t forget the 2π n
on these! There will be problems down
the road in which we will miss solutions without this! Also make sure that it gets put on at this
stage! Now divide by 3 to get all the
critical points for this function.

Notice that in the previous example we got an infinite
number of critical points. That will
happen on occasion so don’t worry about it when it happens.
Example 5 Determine
all the critical points for the function.
Solution
Here’s the derivative for this function.
Now, this looks unpleasant, however with a little
factoring we can clean things up a little as follows,
This function will exist everywhere and so no critical
points will come from that.
Determining where this is zero is easier than it looks. We know that exponentials are never zero
and so the only way the derivative will be zero is if,
We will have two critical points for this function.

Example 6 Determine
all the critical points for the function.
Solution
Before getting the derivative let’s notice that since we
can’t take the log of a negative number or zero we will only be able to look
at .
The derivative is then,
Now, this derivative will not exist if x is a negative number or if ,
but then again neither will the function and so these are not critical
points. Remember that the function
will only exist if and nicely enough the derivative will also
only exist if and so the only thing we need to worry about
is where the derivative is zero.
First note that, despite appearances, the derivative will
not be zero for . As noted above the derivative doesn’t exist
at because of the natural logarithm and so the
derivative can’t be zero there!
So, the derivative will only be zero if,
Recall that we can solve this by exponentiating both
sides.
There is a single critical point for this function.

Let’s work one more problem to make a point.
Example 7 Determine
all the critical points for the function.
Solution
Note that this function is not much different from the
function used in Example 5. In this
case the derivative is,
This function will never be zero for any real value of x.
The exponential is never zero of course and the polynomial will only
be zero if x is complex and recall
that we only want real values of x
for critical points.
Therefore, this function will not have any critical
points.

It is important to note that not all functions will have
critical points! In this course most of
the functions that we will be looking at do have critical points. That is only because those problems make for
more interesting examples. Do not let
this fact lead you to always expect that a function will have critical
points. Sometimes they don’t as this
final example has shown.