= Solution
For $l=0$ put $L_D=NH/f_0$, $\eta=kL_D$ with $k>0$, and write the discriminant as
$$
D(\eta)=\left(\frac\eta2-\coth\frac\eta2\right)\left(\frac\eta2-\tanh\frac\eta2\right)
=1+\frac{\eta^2}{4}-\eta\coth\eta.
$$
The second factor is positive for every $\eta>0$, since $x>\tanh x$. The first changes sign at the positive solution of $x=\coth x$, $x=\eta/2$. Numerically $x_c=1.19968$, so $\eta_c=2.39936$. The growing branch exists for $0<kL_D<\eta_c$, giving
$$
\boxed{\lambda>\lambda_{\min}=\frac{2\pi}{\eta_c}L_D=2.619L_D}.
$$
Equality is neutral, so this is an infimum <wavelength>, rather than an attained shortest growing mode.
In the unstable interval,
$$
\boxed{c_r=\frac{\Lambda H}{2},\qquad
\sigma\equiv kc_i=\frac{\Lambda f_0}{N}\sqrt{-D(\eta)}}.
$$
Both phase-speed branches have the same <real part> there; the imaginary parts have opposite signs. At and beyond the short-wave cutoff the growth rate is zero and the two real speeds are
$$
\frac{c_\pm}{\Lambda H}=\frac12\pm\frac{\sqrt{D(\eta)}}\eta.
$$
They split continuously from $1/2$ and approach the boundary-flow speeds zero and $\Lambda H$ at large $\eta$. As $\eta\to0$, $D=-\eta^2/12+O(\eta^4)$, so $\sigma\to0$ linearly with $k$, despite a finite limiting $c_i=\Lambda H/\sqrt{12}$. A zonally uniform disturbance has no exponential growth from this mechanism.
The maximum growth minimizes $D$. Its derivative is
$$
D'(\eta)=\frac\eta2-\coth\eta+\eta\,\operatorname{csch}^2\eta.
$$
The positive interior minimum is at $\eta_m\simeq1.606$, yielding
$$
\boxed{\lambda_m=\frac{2\pi}{\eta_m}L_D\simeq3.91L_D,\qquad
\sigma_{\max}=0.310\frac{\Lambda f_0}{N}}.
$$
The supplied phase-speed formula therefore predicts fastest growth at a <wavelength> of order four deformation lengths, not at the short-wave cutoff.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2015-baroclinic-phase-speed-growth.png]
{title=Phase speeds and growth rate of the rigid-lid constant-shear baroclinic mode; cutoff and fastest-growing wavelengths follow from the supplied discriminant}
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