Proving the Uniqueness of a Fixed Point for a Differentiable Function

[dancergirlie]

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!

 

[Dick]

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]?

 

[dancergirlie]

Thanks so much for the help!