009A Sample Final 1, Problem 10
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Consider the following continuous function:
defined on the closed, bounded interval .
a) Find all the critical points for .
b) Determine the absolute maximum and absolute minimum values for on the interval .
Foundations: 

Recall: 
1. To find the critical points for we set and solve for 

2. To find the absolute maximum and minimum of on an interval 

Solution:
(a)
Step 1: 

To find the critical points, first we need to find 
Using the Product Rule, we have 

Step 2: 

Notice is undefined when 
Now, we need to set 
So, we get 

We cross multiply to get 
Solving, we get 
Thus, the critical points for are and 
(b)
Step 1: 

We need to compare the values of at the critical points and at the endpoints of the interval. 
Using the equation given, we have and 
Step 2: 

Comparing the values in Step 1 with the critical points in (a), the absolute maximum value for is 
and the absolute minimum value for is 
Final Answer: 

(a) and 
(b) The absolute minimum value for is 