A mass M is broken into two parts m and (M-m). What should be the ratio between m and M so that the force of gravitation between them is maximum ?

If r is the distance between $\mathrm{m}$ and $\mathrm{M}–\mathrm{m}$, the gravitational force will be
$\mathrm{F}=\mathrm{G}\frac{\mathrm{m}(\mathrm{M}-\mathrm{m})}{{\mathrm{r}}^2}=\frac{\mathrm{G}}{{\mathrm{r}}^2}\left(\mathrm{mM}-{\mathrm{m}}^2\right)$
For F to be maximum, $\frac{\mathrm{dF}}{\mathrm{dm}}=0$ (M and r constant )
$\begin{array}{l}\Rightarrow \frac{\mathrm{d}}{\mathrm{dm}}\left[\frac{\mathrm{G}}{{\mathrm{r}}^2}\left(\mathrm{mM}-{\mathrm{m}}^2\right)\right]=0 \\ \Rightarrow \mathrm{M}-2\mathrm{m}=0\\ \Rightarrow \frac{\mathrm{m}}{\mathrm{M}}=\frac{1}{2}\end{array}$