Solution
= Solution
A <fixed point> of $F$ is also a <fixed point> of $F\circ F$. By the <chain rule>,
$$
\boxed{(F\circ F)'(u_*)=F'(F(u_*))F'(u_*)=[F'(u_*)]^2.}
$$
If the original <fixed-point multiplier> has modulus greater than one, its square also exceeds one, so the same <fixed point> is unstable for the twice-iterated map. New stable <fixed points> of the twice-iterated map, if present, must be different points; they can represent a genuine <two-cycle> of the original map.