= Solution
Let $a=v_A^2k^2/\Omega^2$. The growing root of the <ideal magnetorotational dispersion relation> is
$$
\frac{s^2}{\Omega^2}=y(a)=\frac{-(1+2a)+\sqrt{1+16a}}2,\qquad0<a<3.
$$
Its derivative is $y'(a)=-1+4/\sqrt{1+16a}$, so the continuum maximum occurs at $a=15/16$, with $s=3\Omega/4$. For the <bounded vertical modes of a sech-squared magnetized disk>,
$$
a_n=\frac{2n(n+1)}\beta=\frac{n(n+1)}{12}\qquad(\beta=24).
$$
Modes $n=1,\ldots,5$ are unstable; $n\geq6$ have $a_n\geq3$. The function $y$ increases up to $15/16$ and decreases thereafter, so the only candidates for the discrete maximum are $n=2$, with $a_2=1/2$, and $n=3$, with $a_3=1$. Their values are $y(a_2)=1/2$ and $y(a_3)=(\sqrt{17}-3)/2>1/2$. Hence
$$
\boxed{n=3,\qquad s=\Omega\sqrt{\frac{\sqrt{17}-3}{2}}\simeq0.74937\,\Omega.}
$$
The associated velocity profile is $F_3=[5\tanh^3(z/H)-3\tanh(z/H)]/2$. Discrete vertical quantization makes its growth slightly smaller than the continuum maximum. This is the <fastest discrete mode of a stratified magnetorotational instability>.
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