Science:Math Exam Resources/Courses/MATH110/April 2017/Question 11 (a)
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Question 11 (a) 

(a) Carefully prove that the function has at least one critical number . 
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! 
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Hint 

Compute the derivative of , and then apply the intermediate value theorem to to show that there must be some such that 
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.

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Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint, it is sufficient to check that satisfies the assumptions of the intermediate value theorem. First note that is a continuous function on (in fact, it is continuous on the whole domain ). We know that This shows that is a value between and Thus, by the intermediate value theorem, there exists at least one such that That is, is the critical value of . 
Please rate how easy you found this problem:
Hard Easy 
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