Solution
= Solution
Linearization about zero and expansion in the <Fourier sine series> give, for the $n$th mode,
$$
\ddot q_n+\left(f'(0)+\frac{n^2}{R}\right)q_n=0.
$$
Every mode is oscillatory exactly when the smallest coefficient, at $n=1$, is nonnegative. Since $f'(0)<0$,
$$
\boxed{R\leq R_c=-\frac1{f'(0)}}.
$$