Given the function $f(x) = {-x^4 \over 4} + x^3 -4x + 4$ I have graphically localized two roots $\alpha$ and $\beta$ (with $\alpha < \beta$). After analyzing them with Newton's algorithm I'm given the fixed point equation $g(x) = x - {f(x) \over m}$ with $m \ne 0$ and I'm asked to verify that $\alpha$ and $\beta$ are fixed points for $g$.
How can I verify that $\alpha$ and $\beta$ are fixed points of $g$ if I don't have the exact root points to test $g(x) = x$?
As @dxiv pointed out, it's really straight forward. $g(x)=x$ only when the term ${f(x) \over m} = 0$ which is only when $f(x) = 0$ and, since $\alpha$ and $\beta$ are the roots of $f(x)$, when $x=\alpha$ or $x=\beta$ the term $f(x)\over m$ nullifies, resulting whith $g(x)=x$