Total mechanical energy of the body at the surface of earth

$$\begin{gathered} M_i=-\frac{GMm}{R}+\frac{1}{2}m{\left[\frac{1}{3}\left(\sqrt{\frac{2GM}{R}}\right)\right]}^2 \\ =-\frac{GMm}{R}+\frac{GMm}{9R} \\ =-\frac{8GMm}{9R} \end{gathered}$$

Total mechanical energy of the body at the highest point

$M_f=-\frac{GMm}{R+h}$

Using conservation of mechanical energy,

$$\begin{gathered} M_i=M_f \\ \Rightarrow -\frac{GMm}{R+h}=-\frac{8GMm}{9\mathrm{R}} \end{gathered}$$

$\Rightarrow R+h=\frac{9R}{8}$

$\Rightarrow h=\frac{R}{8}=\frac{6400}{8}=800 \mathrm{km}$