Science:Math Exam Resources/Courses/MATH100/December 2015/Question 12 (a)
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Question 12 (a) 

Let be a continuous function for which and exist. If has at least three zeroes, then how many zeroes must and have? Explain your answer carefully. 
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 

Apply Mean Value Theorem to and . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall the mean value theorem: if is differentiable on , then there exists a number such that
Let be the three zeros of the function . i.e., . Note that and are differentiable, so that we can apply this theorem for these functions.
Similarly, applying the theorem for , , and , we can find such that
Therefore, must have at least two zeros.
Note that since , , As a result, must have at least one zero. Therefore, the answer is . 