Thread: 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.

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?

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.

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 were both fixed points then the MVT guarantees a number in between the two where the derivative is one.