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Mean Value Theorem

  1. Mar 2, 2009 #1
    1. The problem statement, all variables and given/known data
    A number a is called a fixed point if f(a)=a. Prove that if f is a differentiable function with f'(x)=1 for all x then f has at most one fixed point.


    2. Relevant equations
    In class we have been using Rolle's Theorem and the Mean Value Theorem.


    3. The attempt at a solution
    In all honest I wasn't sure where to start but this is what i've come up with so far. Knowing that the slope or f'(x)=1 then the original function must have been something like f(x)= x + k. Considering k as a constant that could exist or could not. Then the function either has no fixed point. Or every point of the function is fixed. Therefore giving us a contradiction in the statement. Meaning that this statement cannot be possible. We worked a couple of these in class and I didn't really know how to approach this problem. What I did kinda makes sense to me although it doesn't seem like this should be the answer. Any help would be appreciated thanks!
     
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  3. Mar 2, 2009 #2

    Tom Mattson

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    Let [itex]x=a[/itex] be a fixed point of [itex]f[/itex]. Then by the definition of "fixed point", [itex]f(a)=a+k=a[/itex]. Consider 2 cases: [itex]k=0[/itex] and [itex]k\neq 0[/itex].
     
  4. Mar 2, 2009 #3

    Tom Mattson

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    Hold on a second...The proposition in the problem statement is false. Let [itex]f(x)=x[/itex]. Then [itex]f'(x)=1[/itex] for all [itex]x[/itex], and every point is a fixed point!
     
  5. Mar 2, 2009 #4
    Agreed.
     
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