Science:Math Exam Resources/Courses/MATH110/April 2013/Question 03 (d)
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Question 03 (d) 

Let on the interval . In this question you will sketch this function. (d) Determine where ƒ is concave up and where it is concave down. Does ƒ have any inflection points? 
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? 
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Hint 

Similar to the previous question, now find the second derivative, its possible inflection points, and test the sign of the second derivative between possible inflection points. 
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Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. We know from the previous part of the question that We take the derivative of to find: Setting equal to zero to solve for possible inflection points gives: or, equivalently, Using the unit circle we see that x = π/2 is the only solution in of the equation . This is our only possible point of inflection, because is continuous everywhere. Testing the values of with the second derivative test gives:
Hence has an inflection point at . 