= Solution
{c}
The integral <Gauss's law> is
$$
\int_{\partial V}\mathbf E\cdot\mathbf n\,dS
=\frac{Q(V)}{\epsilon_0}
=\frac1{\epsilon_0}\int_V\rho\,dV.
$$
The <divergence theorem> makes the left side $\int_V\nabla\cdot\mathbf E\,dV$. If the differential relation $\nabla\cdot\mathbf E=\rho/\epsilon_0$ holds, this proves the integral law. Conversely, if the integral identity holds for every suitable volume and the fields are sufficiently regular, a nonzero value of $\nabla\cdot\mathbf E-\rho/\epsilon_0$ would persist in sign on a small neighborhood and contradict its zero integral. The differential law follows.
For a sphere of radius $r$, spherical symmetry makes the flux $4\pi r^2E(r)$ and the enclosed charge $4\pi\int_0^rs^2\rho(s)\,ds$. Hence
$$
\boxed{E(r)=\frac1{\epsilon_0r^2}\int_0^rs^2\rho(s)\,ds.}
$$
There is no extra point charge at the origin in the stipulated density distribution. Outside the charge support,
$$
Q=4\pi\int_0^as^2\rho(s)\,ds,\qquad
\boxed{E(r)=\frac{Q}{4\pi\epsilon_0r^2}\quad(r>a).}
$$
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