Analysis derivative proof

  • #1
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Homework Statement


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.


Homework Equations





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!
 

Answers and Replies

  • #2
Dick
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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]?
 
  • #3
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Thanks so much for the help!!!
 

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