Let r be the direction between the parts
$F= \large\frac{Gm(M-m)}{r^2}$
$\qquad= \large\frac{G}{r^2} $$(mM-m^2)$
The force will be maximum
$\large\frac{dF}{dm}$$=0$
ie $\large\frac{d}{dm} \bigg[\large\frac{G}{r^2} $$(mM-m^2)\bigg]=0$
$\large\frac{G}{r^2}$$(M-2m)=0$
$M-2m=0$
$M=2m$
$\large\frac{M}{m}$$=2$
or $\large\frac{m}{M}=\large\frac{1}{2}$
Hence d is the correct answer.