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 $f'(0)<0$,
$$
\boxed{R\leq R_c=-\frac1{f'(0)}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 2 / ii / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-331/2/ii/a/solution