Monotonic density at fixed chemical potential in a square-root density of states
= Monotonic density at fixed chemical potential in a square-root density of states
The <temperature> derivative of the Fermi occupation has opposite signs below and above $\mu$. Pairing energies $\mu\pm x$ leaves a positive contribution because the <density of states> increases with energy. The unpaired higher-energy tail is also positive. Hence the density increases at fixed $\mu$ for an unbounded three-dimensional parabolic band.