Fixed Points- Proof

• Jan 18th 2010, 05:53 PM
amm345
Fixed Points- Proof
A number a is called a fixed point of a function f if f(a)=a. Assuming that f is differentiable and that for every x one has f'(x)does not equal1 prove that f has at most one fixed point.
• Jan 18th 2010, 06:01 PM
Drexel28
Quote:

Originally Posted by amm345
A number a is called a fixed point of a function f if f(a)=a. Assuming that f is differentiable and that for every x one has f'(x)does not equal1 prove that f has at most one fixed point.

What do you think?
• Jan 18th 2010, 06:23 PM
amm345
I think I've got it, can you check this for me?

Suppose f has two fixed point, x1 <x2. Then by MVT we have that
y (x1,x2),f(x2) f(x1)= f0(y). x2 x1
Then f0(y)=1 which is a contradiction.
• Jan 18th 2010, 06:29 PM
Drexel28
Quote:

Originally Posted by amm345
I think I've got it, can you check this for me?

Suppose f has two fixed point, x1 <x2. Then by MVT we have that
y (x1,x2),f(x2) f(x1)= f0(y). x2 x1
Then f0(y)=1 which is a contradiction.

This is hard to decipher what you mean. Basically all you need to note is that if $x_1\ne x_2$ were both fixed points then the MVT guarantees a number in between the two where the derivative is one.