  We consider convection in an internally heated layer of fluid that is bounded
below by a perfect insulator and above by a poor conductor. The poorly
conducting boundary is modelled by a fixed heat flux. Using solely analytical
methods, we find linear and energy stability thresholds for the static state,
and we construct a lower bound on the mean temperature that applies to all
flows. The linear stability analysis yields a Rayleigh number above which the
static state is linearly unstable ($R_L$), and the energy analysis yields a
Rayleigh number below which it is globally stable ($R_E$). For various boundary
conditions on the velocity, exact expressions for $R_L$ and $R_E$ are found
using long-wavelength asymptotics. Each $R_E$ is strictly smaller than the
corresponding $R_L$ but is within 1%. The lower bound on the mean temperature
is proven for no-slip velocity boundary conditions using the background method.
The bound guarantees that the mean temperature of the fluid, relative to that
of the top boundary, grows with the heating rate ($H$) no slower than
$H^{2/3}$.
