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

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, . (c) Determine the interval(s) where f(x) is concave up and concave down, respectively. You may use:

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 

Use the given second derivative and check the sign of the second derivative between points where the second derivative is zero and undefined. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We are given that
and this second derivative is undefined at . The second derivative is 0 when , that is, when . So we check the intervals individually. Note that for all values of . Case 1: We test that the second derivative at a point in the interval, say , which gives a value of . Thus the function is concave up 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 concave up 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 concave down on this interval. This completes the question. 