Prove: Mean Value Theorem & Rolle's Theorem | At Most 1 Fixed Point

[DeltaIceman]

Homework Statement

A number a is called a fixed point if f(a)=a. Prove that if f is a differentiable function with f'(x)=1 for all x then f has at most one fixed point.

Homework Equations

In class we have been using Rolle's Theorem and the Mean Value Theorem.

The Attempt at a Solution

In all honest I wasn't sure where to start but this is what I've come up with so far. Knowing that the slope or f'(x)=1 then the original function must have been something like f(x)= x + k. Considering k as a constant that could exist or could not. Then the function either has no fixed point. Or every point of the function is fixed. Therefore giving us a contradiction in the statement. Meaning that this statement cannot be possible. We worked a couple of these in class and I didn't really know how to approach this problem. What I did kinda makes sense to me although it doesn't seem like this should be the answer. Any help would be appreciated thanks!

 

[quantumdude]

Let $x=a$ be a fixed point of $f$. Then by the definition of "fixed point", $f(a)=a+k=a$. Consider 2 cases: $k=0$ and $k\neq 0$.

 

[quantumdude]

Hold on a second...The proposition in the problem statement is false. Let $f(x)=x$. Then $f'(x)=1$ for all $x$, and every point is a fixed point!

 

[csprof2000]

Agreed.