Science:Math Exam Resources/Courses/MATH110/April 2011/Question 03 (c)
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Question 03 (c) 

In this question you will state and then apply the Mean Value Theorem. Use your answer in part (b) to explain why we may conclude that for any positive value b. 
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 

What is the greatest value that the derivative of can achieve on the interval ? How does this bound the slope you found in part (b)? 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. In the previous question, we showed that for some in the interval Now we know that for all possible values of , is bounded above by 1, or Substituting what we know from part (b), this is the same as Because we can multiply both sides by b to get Which completes our proof. 