Critical Points
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
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We say that
 is
a critical point of the function f(x) if
 exists
and if either of the following are true.
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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
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,
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