How many fixed points can a differentiable function have?

Question

Let $f$ be a differentiable function on $\mathbb{R}$. Then could any one tell me which of the following statements are necessarily true?

1. If $f'(x)\le r<1$ for all $x$ then $f$ has at least one fixed point

2. If $f$ has a unique fixed point, then $f'(x)\le r<1$ for all $x$

3. If $f'(x)\le r<1$ for all $x$ then it has a unique fixed point.

4. If $f$ has a unique fixed point, then $f'(x)\ge r>-1$ for all $x$

We know that if $X$ is a complete metric space, $f:X\to X$ is a map such that $d(f(x),f(y)\le cd(x,y)$, where $0\le c<1$. Then $f$ has a unique fixed point, here by MVT we get $\lvert f(x)-f(y) \rvert = \lvert f'(c)\rvert \cdot \lvert x-y \rvert$ for any $x,y\in\mathbb{R}$, so $f$ is a contraction map and hence by above theorem $f$ has a unique fixed point so $1$ is true, what can be said more?

Answer 1

Hints:

For 2. consider $f(x)=2x$.

For 3. Let $a$ and $b$ be fixed points of $f$ and consider $\displaystyle\int_a^b f^\prime(x)\,\mathrm{d}x$.

For 4. consider $f(x) = -x$.