Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 13 (d)
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Question 13 (d) 

(d) Is , or ? Determine which of these three options is correct, and justify your answer. 
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? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Use the quadratic approximation with a cubic remainder. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By the Hint, we consider the quadratic approximation with cubic remainder and write where is a number between and .
For , we have where is a number between and .
Let us find the third derivative of . Recalling that , we have . For , because in particular.
Putting everything together, we see that with . Therefore the last term is positive and we conclude that or Answer: we have . 