Fermi distribution — UPSC CSE 2008 Physics PYQ

Show that at T = 0, the Fermi distribution function has a value 1 for energies less than the Fermi energy \epsilon_F and is zero above it. For a system of non-interacting electrons at T = 0, show that the ground state energy of the system of N particles is \frac{3}{5} N \epsilon_F.