Let $$x_{n+1}=x_n(r-x_n)$$ where $r$ is a parameter.
a. Find the fixed points
b. For which values of $r$ the iteration has a converge fixed points? For which fixed points converge in an area of a specific point? If there is a converge what is the order? If not Prove that it does not converge
a. we need to solve $$x=x(r-x)\Rightarrow x-x(r-x)=0\Rightarrow x(1-r+x)=0$$
So the fixed points are: $x=0$ and $x=r-1$
b. I know that there is something which $|g'(x)|<1$ and $|g'(x)|>1$ where $g(x)=x(r-x)$, but there is no interval to check it