Science:Math Exam Resources/Courses/MATH100/December 2013/Question 08 (b)
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Question 08 (b) 

FullSolution Problem. Justify your answer and show your work. Full simplification of numerical answers is required unless explicitly stated otherwise. Let . Determine the intervals where is increasing, and the intervals where is decreasing. (The results from part (a) may be useful.) 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

What is the sign of when is increasing? When is decreasing? 
Hint 2 

We know from part (a) that and that when . How can we use this to find where and where ? 
Hint 3 

Note also that has a vertical asymptote at . What can we say about the signs of when and when ? 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. Where , is increasing; where , is decreasing. From part (a), we note that may change sign at its critical points and . We can construct a sign table to organize our calculations: From this table we observe that is increasing on and decreasing on . 