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

Mass of small element of planet of radius x and thickness 
 $dm=\rho \times 4\pi x^{2}dx={\rho }_0(1-\frac{x^2}{R^2})\times 4\pi x^{2}dx$
Mass of the planet from  $r=0tor$
$M=4\pi {\rho }_0\underset{0}{\overset{r}{\int }}(x^2-\frac{x^4}{R^2})dx$ $\Rightarrow M=4\pi {\rho }_0|\frac{r^3}{3}-\frac{r^5}{5R^2}|$

Gravitation field     $E=\frac{GM}{r^2}=\frac{G}{r^2}\times 4\pi {\rho }_0(\frac{r^3}{3}-\frac{r^5}{5R^2})$ $\Rightarrow E=4\pi G{\rho }_0(\frac{r}{3}-\frac{r^3}{5R^2})$
E is maximum when  $\frac{dE}{dr}=0$

$\begin{array}{l}\Rightarrow \frac{dE}{dr}=4\pi G{\rho }_0(\frac{1}{3}-\frac{3r^2}{5R^2})=0\\ \Rightarrow r=\frac{\sqrt{5}}{3}R\end{array}$