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Math Help - Thank you! (Rolle's Theorem)

  1. #1
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    Thank you! (Rolle's Theorem)

    Hi all,

    I just got back from Calculus final and am so elated. I only missed 2 questions and one of them was the Bonus question which is meant to be harder. It introduced the concept of a fixed point (f(a)=a) and asked us to prove that if f'(x) does not equal 1 for every real number x, then there will be no more than one fixed point

    It gave us a hint: Apply Rolle's Theorem to g(x)= f(x) - x

    I THINK this was how it was laid out but I couldn't get it. Any help is appreciated. And thanks to this forum! Definitely helped me ace this test
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  2. #2
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    Suppose there are 2 fixed points a and b.
    g(a) = 0 and g(b) = 0
    so g'(x) = 0 somewhere between a and b by Rolle's Theorem.

    Can you arrive at a contradiction?
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  3. #3
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    Quote Originally Posted by DannyMath View Post
    Hi all,

    I just got back from Calculus final and am so elated. I only missed 2 questions and one of them was the Bonus question which is meant to be harder. It introduced the concept of a fixed point (f(a)=a) and asked us to prove that if f'(x) does not equal 1 for every real number x
    Surely you meant to say "does not equal 1 for any real number x"

    , then there will be no more than one fixed point

    It gave us a hint: Apply Rolle's Theorem to g(x)= f(x) - x

    I THINK this was how it was laid out but I couldn't get it. Any help is appreciated. And thanks to this forum! Definitely helped me ace this test
    Follow Math Help Forum on Facebook and Google+

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