Question 03 (b)
Consider a function ƒ whose derivative is equal to 0 on the entire interval . Let x1 and x2 be two points on . Prove, using your answer in part (a), that
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!
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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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,