Given that $\displaystyle f$ is function that's differentiable on the interval $\displaystyle [a,b]$ where $\displaystyle f'(x) \neq 1\,\,\, \forall x \in [a,b]$.

1.) Prove $\displaystyle f$ has at most 1 fixed point on the interval $\displaystyle [a,b]$. (ie., show there's at most 1 point $\displaystyle x \in [a,b]$ where $\displaystyle f(x) = x$

2.) By giving specific examples, show it's possible that $\displaystyle f$ either has no fixed points or that $\displaystyle f$ has 1 fixed point on $\displaystyle [a,b]$.