Science:Math Exam Resources/Courses/MATH100/December 2011/Question 05 (d)
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Question 05 (d) 

FullSolution Problems. In questions 28, justify your answers and show all your work. Simplification of answers is not required unless explicitly stated. Let
Find the intervals on which y=f(x) is concave up. 
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 

We already computed the second derivative in part (c). Checking where this is positive tells you when your function is concave up. 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. From part (c), we know that Notice that the potential inflection points are at x=0,4, where is either zero or undefined. Doing a sign chart for this function yields the following. When x < 0 we have that . When we have that . When we have that . Hence, the function is concave up on . Note: Since the sign of changes at both points, both and are inflection points. Even though is not defined at . 