# Thank you! (Rolle's Theorem)

• Dec 15th 2010, 08:27 PM
DannyMath
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 :)
• Dec 15th 2010, 08:32 PM
snowtea
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?
• Dec 16th 2010, 03:53 AM
HallsofIvy
Quote:

Originally Posted by DannyMath
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"

Quote:

, 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 :)