Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 11 (d)
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Question 11 (d) 

The function is an example of a Gamma distribution, a function that may be used to describe certain probabilities.
(d) The second derivative of is Identify all the intervals where is concave up, and all the intervals where is concave down.

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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 

Recall that a function is concave up if its second derivative is positive, and is concave down if its second derivative is negative. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We proceed similarly as in part (c). By the Hint, we consider the sign of , which is the same as the sign of Clearly, this is positive when and is negative when .
Therefore, is concave up on the interval where , that is on . Similarly, is concave down on the interval where , or . Answer: is and . 