= Solution
For the high-frequency indices $(n+1)/2\leq k\leq n$, the angle $\theta_k=k\pi/(n+1)$ ranges from $\pi/2$ to $\pi-\pi/(n+1)$. Since
$$
\lambda_k(\omega)=1-\omega+\omega\cos\theta_k,
$$
the large-$n$ <high-frequency smoothing factor of weighted Jacobi> is
$$
\boxed{
\mu_\omega
=\max_{\pi/2\leq\theta\leq\pi}
|1-\omega+\omega\cos\theta|
=\max\{|1-\omega|,|1-2\omega|\}.}
$$
For $0<\omega<1$, this is strictly less than one independently of $n$, so high-frequency error decays geometrically much faster than the low-frequency factor $1-O(n^{-2})$.
To minimize $\mu_\omega$, balance the endpoint magnitudes:
$$
1-\omega=2\omega-1.
$$
Thus the optimal relaxation and attenuation factors are
$$
\boxed{\omega_*=\frac23,
\qquad
\mu_*=\frac13.}
$$
For finite $n$, retaining the endpoint $\cos(\pi-\pi/(n+1))=-\cos(\pi/(n+1))$ gives
$$
\omega_*=\frac{2}{2+\cos(\pi/(n+1))},
\qquad
\mu_*=\frac{\cos(\pi/(n+1))}
{2+\cos(\pi/(n+1))},
$$
which tend to $2/3$ and $1/3$.
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