= Solution
For a period-$n$ orbit of $g(x)=\mu-x^2$ that avoids zero, the <chain rule> gives its <multiplier of a periodic orbit of an iteration> as
$$
(g^n)'(\hat x)=\prod_{k=1}^n g'(x_k)=(-2)^n\prod_{k=1}^n x_k.
$$
The sign of $g'$ is positive on the negative half-axis and negative on the positive half-axis, exactly as the parabola sketch shows. Consequently
$$
\boxed{\operatorname{sgn}((g^n)'(\hat x))=(-1)^n\prod_{k=1}^n\operatorname{sgn}(x_k).}
$$
This product is independent of the chosen starting point because changing $\hat x$ only cyclically reorders the factors. If the orbit contains zero, the derivative product is zero and its orientation is undefined. One must then use the actual zero <periodic-orbit multiplier>, rather than infer a positive or negative orientation from the special assignment $\operatorname{sgn}(0)=1$. The nonzero-<periodic-orbit multiplier> cases needed in part (d) avoid this exception automatically.
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