Question

for a three-dimensional free-electron metal, at what fraction of the fermi energy will the density of occupied states be a maximum for

Answer 1

for \[k_B T=0.1E_F \]
where \[E_F\] is the fermi energy

Answer 2

Do you mean the maximum of the density of states multiplied by the Fermi-Dirac distribution function ?
I differentiated that product with respect to energy and ended up with an equation that could be solved graphically, but it assumes that you know what the chemical potential is.
Probably there is a smarter way to get an answer, but I don't know it.