The mass density of a planet of radius $R$  varies with the distance   $r$ from its centre as $\rho \left(r\right)={\rho }_0\left(1-\frac{r^2}{R^2}\right)$   . Then the gravitational field is maximum at:

$E=\frac{G{m}_{\text{enc}}}{r^2}=\frac{G}{r^2}{\int }_0^r?{\rho }_0\left(1-\frac{r^2}{R^2}\right)4\pi r^{2}dr$

$\begin{array}{rr}E& =\frac{G}{r^2}4\pi {\rho }_0{\left[\frac{r^3}{3}-\frac{r^5}{5R^2}\right]}_0^r\\ & =4\pi G{\rho }_0\left[\frac{r}{3}-\frac{r^3}{5R^2}\right]\end{array}$

For $E$  to be $\mathrm{m}\mathrm{a}\mathrm{x},\frac{\mathrm{d}\mathrm{E}}{\mathrm{d}\mathrm{r}}=0$

$\begin{array}{c}\Rightarrow \mathrm{}\frac{1}{3}-\frac{3r^2}{5R^2}=0\\ r=\sqrt[]{\frac{5}{9}}R\end{array}$