= Solution
Count wavevectors in a large volume with <periodic boundary conditions>. One spin-degenerate spherical band supplies
$$
\frac{N(k)}{\mathcal V}=\frac{2}{(2\pi)^3}\frac{4\pi k^3}{3}=\frac{k^3}{3\pi^2},\qquad
\frac1{\mathcal V}\frac{dN}{dk}=\frac{k^2}{\pi^2}.
$$
For the <conduction band>, $k=\sqrt{2m_e^*(\epsilon-\epsilon_c)}/\hbar$ and $dk/d\epsilon=m_e^*/(\hbar^2k)$. Transforming the state count to energy gives
$$
\boxed{g_e(\epsilon)=\frac1{2\pi^2}\left(\frac{2m_e^*}{\hbar^2}\right)^{3/2}\sqrt{\epsilon-\epsilon_c},\qquad\epsilon\geq\epsilon_c.}
$$
For the <valence band> use $|dk/d\epsilon|=m_h^*/(\hbar^2k)$ and $k=\sqrt{2m_h^*(\epsilon_v-\epsilon)}/\hbar$, obtaining
$$
\boxed{g_h(\epsilon)=\frac1{2\pi^2}\left(\frac{2m_h^*}{\hbar^2}\right)^{3/2}\sqrt{\epsilon_v-\epsilon},\qquad\epsilon\leq\epsilon_v.}
$$
Each <density of states> is zero beyond its indicated band edge. The factor two counts spin; an extra valley degeneracy was not assumed in these stipulated isotropic bands.
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