= Solution
In the deep-water limit $kh\gg1$, $\tanh(kh)\sim1$. Put $\beta=B/\rho>0$. The squared <phase speed> of the <flexural-gravity wave> is then
$$
c_p^2\sim\frac gk+\beta k^3.
$$
It diverges as $k\to0^+$ and as $k\to\infty$ in this limiting model. Its unique stationary point satisfies
$$
-\frac g{k^2}+3\beta k^2=0,
\qquad \boxed{k_* =\left(\frac{\rho g}{3B}\right)^{1/4}}.
$$
The second derivative $2g/k^3+6\beta k$ is positive. Using $g/k_*=3\beta k_*^3$ gives the <minimum phase speed of a flexural-gravity wave>:
$$
c_{\min}^2=4\beta k_*^3
=\frac4{3^{3/4}}\beta^{1/4}g^{3/4},
\qquad
\boxed{c_{\min}=\frac2{3^{3/8}}
\left(\frac{Bg^3}\rho\right)^{1/8}}.
$$
This is the deep-water minimum: applying it to a finite depth requires $k_*h\gg1$. Otherwise the $\tanh(kh)$ factor must be retained in the minimization. Within the deep-water approximation, <phase speed> and <group velocity> coincide at $k_*$, although the wave family remains dispersive.
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