A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?
F = G m (M -m)/r^2
maximize
m(M-m) where M is a constant total mass
y = m(M-m) = M m - m^2
dy/dm = M - 2 m
that = 0 when
M = 2 m
m = M/2
so split it in two