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Analysis derivative proof

  1. Nov 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f : (a, b)--> R be differentiable on (a,b), and assume that f'(x) unequal 1 for all x in (a,b).
    Show that there is at most one point c in (a,b) satsifying f(c) = c.


    2. Relevant equations



    3. The attempt at a solution

    I think that We need to use the mean value theorem for this problem.

    This is what I know, but I'm not sure what I should use for my proof:
    If we let c be n (a,b)
    By definition f'(c)=lim: x-->c (f(x)-f(c)/x-c) assuming that f(x) is differentiable at c.

    Also, the mean value theorem states that f[a,b]-->R is continuous on [a.b] and differentiable on (a,b), then there exists a point c in (a,b) so that f'(c)=f(b)-f(a)/b-a

    Any help/hints would be great!
     
  2. jcsd
  3. Nov 27, 2009 #2

    Dick

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    Homework Helper

    Use a proof by contradiction. Suppose there are two points d and e in (a,b), where f(d)=d and f(e)=e. Then what does the mean value theorem tell you if you apply it on the interval [d,e]?
     
  4. Nov 27, 2009 #3
    Thanks so much for the help!!!
     
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