= Solution
Assume a homogeneous <ideal gas>, constant <specific-heat ratio> $\gamma>1$, and an upward displacement that is an <adiabatic process> and maintains pressure balance with its surroundings. If the ambient <temperature> falls faster with decreasing <pressure> than the parcel's <adiabatic temperature gradient>, the parcel becomes hotter and less dense than its surroundings; <buoyancy> amplifies the displacement. This is the <Schwarzschild criterion>:
$$
\nabla\equiv\frac{d\log T}{d\log P}>\nabla_{\rm ad}
=\frac{\gamma-1}{\gamma}\equiv A.
$$
The <square-root exponential atmospheric profile> has
$$
\boxed{\nabla=\frac{2\sqrt{T-T_0}}{\alpha T};\qquad
\text{instability requires }\frac{2\sqrt{T-T_0}}{\alpha T}>A.}
$$
An <atmospheric thermal inversion> with $\alpha<0$ has $\nabla<0$ and is stable to these ordinary adiabatic displacements. For $\alpha>0$, put $s=\sqrt{T-T_0}$. The condition becomes the <quadratic inequality>
$$
A\alpha s^2-2s+A\alpha T_0<0,
$$
intersected with $s\ge0$ and $T_0+s^2>0$. In the usual case $T_0>0$, the maximum gradient occurs at $s=\sqrt{T_0}$, namely $T=2T_0$, and is $1/(\alpha\sqrt{T_0})$. Hence a convectively unstable interval exists precisely when
$$
\boxed{A\alpha\sqrt{T_0}<1,\qquad
s_-<\sqrt{T-T_0}<s_+,\quad
s_\pm=\frac{1\pm\sqrt{1-(A\alpha)^2T_0}}{A\alpha}.}
$$
Equality in the existence condition gives a single neutrally stable point. A molecular-hydrogen-dominated <hot Jupiter> with rotational modes active but vibrational excitation and dissociation negligible has approximately $\gamma=7/5$, so $A=2/7$. The existence condition is then $\alpha\sqrt{T_0}<7/2$. For $T_0\le0$ the general inequality above remains valid, but its roots must be intersected with the positive-<temperature> domain; the positive-$T_0$ maximum formula must not be reused. Composition gradients require the <Ledoux criterion>, and dissociation or variable heat capacities change $A$. Sustained <convection> normally adjusts a superadiabatic profile toward an <adiabatic temperature gradient>.
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