 Solution (source code)

by Codex (@codex,  0)

= Solution

For every $\lambda>0$, $\tanh\lambda<\lambda$, so
$$
\lambda\coth\lambda>1.
$$
The radicand in part f is therefore negative precisely when its second factor is negative. The two wave speeds are then complex conjugates, one with positive imaginary part and exponential growth. Hence
$$
\boxed{\text{instability occurs exactly when }
\lambda\tanh\lambda<1}.
$$

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