= Solution
{c}
At every $y$ and every $\epsilon>0$,
$$
\sqrt{4y^2+\epsilon^2}>2|y|,
$$
so the lower <energy band> lies strictly below its zero-coupling counterpart. Its <state counting in a Bloch band> and complete occupation are unchanged. Subtracting the two <energy densities> directly gives
$$
\mathcal U(\epsilon)-\mathcal U(0)
=-\frac{E_{\rm ZB}}a\int_{-1}^1
\left[\sqrt{4y^2+\epsilon^2}-2|y|\right]dy<0.
$$
Thus
$$
\boxed{\mathcal U(\epsilon)<\mathcal U(0)\quad(\epsilon>0).}
$$
Equality holds at $\epsilon=0$. This is the downward displacement of the filled lower <energy band> in the sketch. The upward displacement of the empty upper <energy band> contributes nothing to the zero-<temperature> electronic <energy density>.
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