= Solution
Taking $E_1>0$ so that the zone centre is a minimum, the <effective mass for exponentially decaying tight-binding hopping> is
$$
\boxed{m^*(a)=\frac{\hbar^2}{4E_1a^2}e^{a/a_1}.}
$$
Its logarithmic derivative is
$$
\frac1{m^*}\frac{dm^*}{da}=\frac1{a_1}-\frac2a.
$$
The curve decreases for $a<2a_1$, has a unique minimum at $a=2a_1$, and increases for $a>2a_1$:
$$
\boxed{m^*_{\min}=\frac{\hbar^2\exp(2)}{16E_1a_1^2}.}
$$
The formal limits are an $a^{-2}$ divergence as $a\to0$ and an exponential divergence as $a\to\infty$. The small-separation extrapolation eventually invalidates the localized-orbital <tight-binding model>; the sketch describes the specified formula.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2021-q4-effective-mass.png]
{title=Effective mass versus atomic spacing for exponentially decaying nearest-neighbour hopping}
{description=Original plot of the specified tight-binding model, normalized by its minimum. The minimum occurs at $a=2a_1$, where the logarithmic derivative of the hopping magnitude balances the explicit $a^2$ band-curvature factor.}
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