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Question 05 (d) 

Let . Find the intervals on which is concave upward. 
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Hint 

Find the intervals on which the second derivative exists and is nonzero. 
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Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that when , is concave up; likewise, when , is concave down. So what we need to do is find the intervals on which exists and is nonzero. From above, we know that , and indeed, if and only if or . Since for all , we have that . Using the quadratic formula, we find that this quadratic equation has no real roots. Therefore, there is no where . Now, we make a table as before for , noting that does not exist at . This gives us two intervals, namely and where is nonzero and defined:
