Science:Math Exam Resources/Courses/MATH110/April 2012/Question 03 (b)
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Question 03 (b) 

Consider a function ƒ whose derivative is equal to 0 on the entire interval . Let x_{1} and x_{2} be two points on . Prove, using your answer in part (a), that 
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Hint 

Does satisfy the conditions of Mean Value Theorem? What interval would you want to use to apply the Mean Value theorem? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. If , the assumption is obviously true, so suppose . Since everywhere, we can deduce that is differentiable everywhere, and hence also continuous everywhere, in particular on the interval . We can then apply the Mean Value Theorem, which tells us that there exists some c in the interval such that
However, we know everywhere, so and we have and by cross multiplication,
