009A Sample Final 2, Problem 10

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Let

(a) Find all local maximum and local minimum values of    find all intervals where    is increasing and all intervals where    is decreasing.

(b) Find all inflection points of the function    find all intervals where the function    is concave upward and all intervals where    is concave downward.

(c) Find all horizontal asymptotes of the graph  

(d) Sketch the graph of  

Foundations:  
1.   is increasing when    and    is decreasing when  
2. The First Derivative Test tells us when we have a local maximum or local minimum.
3.   is concave up when    and    is concave down when  
4. Inflection points occur when  


Solution:

(a)

Step 1:  
We start by taking the derivative of   
Using the Quotient Rule, we have
       
Now, we set   
So, we have
       
Hence, we have    and  
So, these values of    break up the number line into 3 intervals:
       
Step 2:  
To check whether the function is increasing or decreasing in these intervals, we use testpoints.
For  
For  
For  
Thus,    is increasing on    and decreasing on  
Step 3:  
Using the First Derivative Test,    has a local minimum at    and a local maximum at   
Thus, the local maximum and local minimum values of    are
       

(b)

Step 1:  
To find the intervals when the function is concave up or concave down, we need to find  
Using the Quotient Rule and Chain Rule, we have
       
We set  
So, we have
        
Hence,
       
This value breaks up the number line into four intervals:
       
Step 2:  
Again, we use test points in these four intervals.
For    we have  
For    we have  
For    we have  
For    we have  
Thus,    is concave up on    and concave down on  
Step 3:  
The inflection points occur at
       
Plugging these into   we get the inflection points
       

(c)

Step 1:  
By L'Hopital's Rule, we have
       
Similarly, we have
       
Step 2:  
Since
       
  has a horizontal asymptote
       
(d):  
Insert sketch


Final Answer:  
   (a)      is increasing on    and decreasing on  
           The local maximum value of    is    and the local minimum value of    is   
   (b)      is concave up on    and concave down on  
            The inflection points are  
   (c)   
   (d)    See above

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