Science:Math Exam Resources/Courses/MATH100/December 2016/Question 14 (b)
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Question 14 (b) 

Let be a function so that
Show that has at least one zero in the open interval . 
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 

Using the mean value theorem, show that the second derivative of must be negative somewhere between and . Also, the second derivative must be positive somewhere between and . 
Hint 2 

Apply the intermediate value theorem to the second derivative of and find a zero of . 
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. In the similar way in part (a), we observe that This implies that the derivative must be positive somewhere between and and then negative somewhere between and . This is because by the mean value theorem, we have and such that
and
Thus the second derivative must be negative somewhere between and ; we can find such that
Similarly, the derivative must be negative somewhere between and and then positive between and . Hence the second derivative must be positive somewhere between and . Thus, by the intermediate value theorem, the second derivative has a zero (and changes sign at that zero) somewhere between and . 