= Solution
Let $A=k^2v_A^2>0$ and $C=N^2+\Omega^2$. Solving the <radially stratified magnetorotational dispersion relation> as a quadratic in $s=\sigma^2$ gives
$$
s_\pm=-A-\frac C2\pm\frac12\sqrt{C^2+16\Omega^2A}.
$$
Both roots are real. If $A+N^2-3\Omega^2<0$, their product is negative, so exactly one root is positive and gives exponential growth. Conversely, when $A+N^2-3\Omega^2\geq0$, the coefficient $2A+N^2+\Omega^2\geq A+4\Omega^2>0$, so both roots are nonpositive. Thus \b[the instability criterion for a specified nonzero <wavenumber>] is
$$
\boxed{0<k^2v_A^2<3\Omega^2-N^2.}
$$
In a local continuum of allowed <wavenumbers>, some unstable mode exists exactly when $N^2<3\Omega^2$. A finite disk only permits <wavelengths> fitting its vertical boundaries, so this existence statement also requires an allowed mode in the interval.
The growing root is $s_+$, whose derivative and curvature are
$$
\frac{ds_+}{dA}=-1+\frac{4\Omega^2}{\sqrt{C^2+16\Omega^2A}},\qquad
\frac{d^2s_+}{dA^2}=-\frac{32\Omega^4}{(C^2+16\Omega^2A)^{3/2}}<0.
$$
Setting the derivative to zero gives \b[the fastest-growing interior mode]:
$$
\boxed{k_{\max}^2v_A^2=\Omega^2-\frac{(N^2+\Omega^2)^2}{16\Omega^2}.}
$$
For a real, nonzero <wavenumber>, this expression requires $|N^2+\Omega^2|<4\Omega^2$, equivalently
$$
\boxed{-5\Omega^2<N^2<3\Omega^2.}
$$
Nonnegative right-hand side permits the endpoints, but there the stationary point is at $k=0$, outside the nonzero-<wavenumber> mode used above. Substituting the interior maximizing value gives
$$
s_{\max}=\frac{(3\Omega^2-N^2)^2}{16\Omega^2},\qquad\boxed{\sigma_{\max}=\frac{3\Omega^2-N^2}{4\Omega}.}
$$
This is the <maximum growth rate of radially stratified magnetorotational instability>, taking $\Omega>0$. For $N^2=0$ it reduces to the usual $3\Omega/4$, at $k^2v_A^2=15\Omega^2/16$. Negative $N^2$ enhances growth and extends the unstable band; positive $N^2$ suppresses growth and eventually eliminates it at $N^2\geq3\Omega^2$. <Magnetic tension> enables angular-momentum exchange between displaced parcels, weakening the rotational stabilization that protected the adverse hydrodynamic stratification.
If $N^2\leq-5\Omega^2$, the interior formula is no longer the physical maximum: $s_+$ decreases for $A>0$, and its supremum as $k\to0$ is $-N^2-\Omega^2$. The corresponding limiting growth rate is $\sqrt{-N^2-\Omega^2}$, dominated by the already unstable hydrodynamic branch. At $N^2=-5\Omega^2$ this joins continuously to $2\Omega$. For the smooth <thin disk> estimate $N^2\sim-(H/r)^2\Omega^2$, the ordinary interior maximum applies and its enhancement over $3\Omega/4$ is only of fractional order $(H/r)^2$.
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