I am working on the following problem:
Consider the function
$f: [-1,1] \mapsto \mathbb{R}, f(x) = x(x-1)(x+1)$
What are the fixed points of $f$? What are the fixed points if we extend the domain to $[-2,2]$?
I would be able to show the existence of the fixed point(s) with the Intermediate Value Theorem but I don't know how to find the specific values for where the fixed points are. How does one do this?
@hardmath pointed out in his comment that finding the fixed points of $f$ is related to finding the roots of $g(x)=f(x)-x=x^3-2x$. We see immediately that $x=0$ is a fixed point. Any other fixed points hence must satisfy $$x^2=2\quad\text{or, equivalently}\quad x=\pm\sqrt 2.$$ $\pm\sqrt 2$ are fixed points if and only if they are in the domain of $f$.