Science:Math Exam Resources/Courses/MATH110/April 2016/Question 09 (b)
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Question 09 (b) 

Suppose that the function is differentiable for all and that and . (b) Suppose also that everywhere. Is it possible that ? Explain why or why not. 
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 

Note that , and try to imagine a function that does satisfy everywhere and . To determine whether such a function exists, sketch an example of a differentiable function that satisfies , , and , then try to use the Mean Value Theorem (as many times as you need to) to explain why your drawings end up the way they do. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. No such function exists. To see this, use the Mean Value Theorem twice. Let and . Because everywhere, and are differentiable everywhere and we can use the Mean Value Theorem for and on the closed interval . By the Mean Value Theorem applied to on , there is a number such that and . Now let and , where is as defined above. By the Mean Value Theorem applied to on , there is a number such that and . But then, contradicting the assumption that everywhere. Hence, no such function exists. 