Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 14
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Question 14 

Let . Find all intervals where is concave up. 
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Hint 

Recall that the second derivative tells you information about the concavity of a function. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall how the second derivative gives us information about the concavity of a function: a function is concave up wherever the second derivative is positive and concave down wherever the second derivative is negative. Thus the concavity changes at the points where the second derivative vanishes. First, let’s compute some derivatives: Now, set the second derivative to zero and solve for the points where the concavity changes: So, we now have three intervals: , , and , and we need to determine in which ones the function is concave up. We can do this by evaluating the second derivative at a point in the interval to see if it’s positive or negative. , , and , so we can conclude that is concave up over the intervals and . 