Answer
$\frac{m}{M}=\frac{1}{2}$
Work Step by Step
Step-1: Let us assume that the two parts are separated by a distance of $R$ units. Thus, gravitational force, $$F=G\frac{m(M-m)}{R^2}=G\frac{(Mm-m^2)}{R^2}$$
Step-2: We need to find the $m/M$ ratio for which $F$ is maximum. Let us use calculus to evaluate this. Since $M$ is fixed let us differentiate $F$ with respect to $m$, $$\frac{dF}{dm}=\bigg(\frac{G}{R^2}\bigg)(M-2m) ...... (i)$$For critical points, $$\frac{dF}{dm}=0=\bigg(\frac{G}{R^2}\bigg)(M-2m)$$$$\implies M=2m$$
Step-3: Differentiating $(i)$, we get, $$\frac{d^2F}{dm^2}=-2\bigg(\frac{G}{R^2}\bigg) < 0$$which clearly proves that gravitational force is maximum when $M=2m$.
Step-4: Therefore, $\frac{m}{M}=\frac{m}{2m}=\frac{1}{2}$
