= Solution
At zero temperature, states are occupied up to $E_{\mathrm F}$. Write the <density of states> as $g(E)=C\sqrt E$, including any fixed volume and spin factors in $C$. Then
$$
N=\int_0^{E_{\mathrm F}}C E^{1/2}\,\mathrm dE
=\frac{2C}{3}E_{\mathrm F}^{3/2},
$$
while
$$
U=\int_0^{E_{\mathrm F}}E\,g(E)\,\mathrm dE
=\frac{2C}{5}E_{\mathrm F}^{5/2}.
$$
Dividing eliminates $C$:
$$
\boxed{\langle E\rangle=\frac UN=\frac35E_{\mathrm F}.}
$$
The mean is positive even at absolute zero because the <Pauli exclusion principle> fills a range of momenta; electrons cannot all occupy the zero-momentum state.
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