Science:Math Exam Resources/Courses/MATH100/December 2012/Question 06 (a)
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Question 06 (a) 

FullSolution Problems. In questions 511, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions. 6. Let . Note that the domain of ƒ is the set of all nonzero real numbers; for example, . (a) Determine the interval(s) where is increasing, and the interval(s) where is decreasing. 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Differentiate and check the sign of the derivative between critical points. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Differentiating gives . Thus the critical numbers are and (notice here our derivative and our function are both undefined). So we check the intervals individually. Case 1: We test that the derivative at a point in the interval, say , which gives a value of . Thus the function is increasing on this interval. Case 2: We test that the derivative at a point in the interval, say , which gives a value of . Thus the function is decreasing on this interval. Case 3: We test that the derivative at a point in the interval, say , which gives a value of . Thus the function is increasing on this interval.
Overall, the function in increasing on and decreasing on . 