Science:Math Exam Resources/Courses/MATH104/December 2013/Question 01 (e)
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Question 01 (e) 

Find the slope of the tangent line to at its point of inflection. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

This is a multistep question. How are inflection points related to the second derivative? 
Hint 2 

Inflection points may occur where the second derivative is zero. 
Hint 3 

The slope of the tangent line at the inflection point is the value of the first derivative at that point. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Following the hint, we begin differentiating.
Solving for gives and so . A quick check shows that the sign of the second derivative changes at this point and so we know that is indeed an inflection point. Plugging into the first derivative gives us the slope of the tangent line at the inflection point:
