5.3.7 Let $P$ be a point on the circumcircle of a regular $n$-gon, and let $f(P)$ be the product of the distances from $P$ to each vertex. Find the maximum value of $f(P)$.
Consider a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, with center $O$ and radius 1. Establish a complex plane with $O$ as the origin and $O A_{n}$ as the positive direction of the real axis. Let $P$ be a point on the unit circle, corresponding to the complex number $z$. Then,
$$
f(P)=\prod_{i=1}^{n}\left|z-\varepsilon_{i}\right|=\left|\prod_{i=1}^{n}\left(z-\varepsilon_{1}\right)\right|=\left|z^{n}-1\right|,
$$
Clearly, when $z^{n}=-1$, $f(P)$ reaches its maximum value of 2. At this time, $z=\mathrm{e}^{\mathrm{i} \frac{2k+1}{n} \pi}$ for $k=0,1, \cdots, n-1$, which corresponds to the midpoint of $\overline{A_{i} A_{i+1}}$ (with $A_{n+1}=A_{n}$).