Science:Math Exam Resources/Courses/MATH110/April 2018/Question 03 (c)
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Question 03 (c) 

Consider a function whose first derivative is . Find the coordinates of all inflection points of , if they exist. Make sure you state why they are 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 

Find the roots of . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The first step is to find . We do this using the chain rule. Write , where and . Then the chain rule with and gives Next, we find the roots of by setting and solving for . If , then either or . In the latter case, we obtain two solutions: and . We have found three possible inflection points. Note that we are not quite done yet: not every point at which the second derivative is zero is necessarily an inflection point. For each of the three solutions we found, we need to check whether the second derivative of changes sign at these points. Since the solutions are isolated, it suffices to check the sign of the second derivative at one point on either side of each of the solutions. We have This shows that each of the three points are indeed inflection points.
Answer: The inflection points are . 