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Critical Points

# Critical Points - Critical Points Critical points will show...

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

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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.

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