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 .
Also, we include the values of where is undefined.
2. To find the absolute maximum and minimum of on an interval ,
we need to compare the values of our critical points with and .

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 .

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