= Solution
For two <electrons> per primitive cell,
$$
\boxed{\epsilon_F=\frac{\hbar^2k_F^2}{2m_e}
=\frac{2\pi\hbar^2}{m_ea^2}
\simeq2.99\ \mathrm{eV}}.
$$
Set $A=\hbar^2\pi^2/(2m_ea^2)=2.350\ \mathrm{eV}$. In the <indirect gap in a uniformly split zone-boundary model> approximation stipulated here, the free boundary <energies> are $A$ along $[10]$ at $X=(\pi/a,0)$ and $2A$ along $[11]$ at $M=(\pi/a,\pi/a)$. Opening a symmetric gap gives
$$
\boxed{E_{X,\pm}=A\pm E_g/2=1.85,\ 2.85\ \mathrm{eV}},
\qquad
\boxed{E_{M,\pm}=2A\pm E_g/2=4.20,\ 5.20\ \mathrm{eV}}.
$$
The branches bend away from the free-electron crossings and flatten at their boundary extrema. Importantly, the upper band near $X$ lies below the lower band's corner <energy>:
$$
\boxed{E_{\rm indirect}\simeq E_{X,+}-E_{M,-}=E_g-A=-1.35\ \mathrm{eV}<0}.
$$
Therefore \b[the crystal is metallic despite its positive boundary gaps]; <electron> and <hole> occupations redistribute across overlapping bands. <positive direct gaps can coexist with band overlap>.
A microscopic square-lattice corner has additional plane-wave degeneracy. The quoted symmetric boundary numbers use the question's directional gap model, not an unspecified four-state corner Hamiltonian. For the <indirect gap of a separable square cosine potential>, a separable weak potential with the same face gap gives the lowest corner <energy> $2A-E_g=3.70\ \mathrm{eV}$, still above the $2.85\ \mathrm{eV}$ upper-band face <energy>. This refinement preserves the metallic conclusion.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2014/paper-2-square-bands.png]
{title=Original directional avoided crossings and the overlap of lower- and upper-band boundary energies}
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