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

For points  inside the planet, at distance r from the center $\left(r<R\right)$,  g is given by 

$\begin{array}{l}E=\frac{G M_{enclosed}}{r^2}\\ here M_{enclosed}={\int }_0^r\rho dV={\int }_0^r{\rho }_0\left(1-\frac{r^2}{R^2}\right)4\pi r^{2}dr\\ \Rightarrow E=\frac{G{\int }_0^r{\rho }_0\left(1-\frac{r^2}{R^2}\right)4\pi r^{2}dr}{r^2}\\ \Rightarrow E=G{\rho }_{0}4\pi \left(\frac{r}{3}-\frac{r^3}{5R^2}\right)\\ \text{ For }\max ima, \frac{dE}{dr}=0\\ \Rightarrow \frac{d}{dr}G{\rho }_{0}4\pi \left(\frac{r}{3}-\frac{r^3}{5R^2}\right)=0\\ \Rightarrow \frac{d}{dr}\left(\frac{r}{3}-\frac{r^3}{5R^2}\right)=0\\ \therefore \frac{1}{3}-\frac{3r^2}{5R^2}=0\\ \Rightarrow r=\sqrt{\frac{5}{9}}R\end{array}$