Science:Math Exam Resources/Courses/MATH110/April 2018/Question 07 (e)
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Question 07 (e) 

Let . (e) Given that
and the fact that for all , find the intervals where is concave up and the intervals where it is concave down.

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Hint 

If on an interval , then is concave up on the interval. On the other hand, if on an interval , then is concave down on the interval. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since and for any , the sign of the numerator in is always positive.
In other words, the sign of the second derivative is determined by that of the denominator. Since when and when , we have
Namely, is concave down on , while it is concave up on .
Answer: 