Explanation

(d) If $r$ is the distance between $q$ and ($\mathcal{Q -}q$), the Coulomb force between them is
$F = \frac{1}{4\pi\varepsilon_{0}}.\frac{q(\mathcal{Q -}q)}{r^{2}}$
$= \frac{1}{4\pi\varepsilon_{0}}.\frac{(q\mathcal{Q -}q^{2})}{r^{2}}$
For $F$ to be maximum $dF/dq$=0 and $\mathcal{Q}$ and $r$ are constants,
$ie$ $\frac{d}{dq}\left\lbrack \frac{1}{4\pi\varepsilon_{0}}\left( q\mathcal{Q -}q^{2} \right) \right\rbrack = 0$
$i.e,$ $\mathcal{Q -}2_{q} = 0$ $\left( as\frac{1}{4\pi\varepsilon_{0}} \neq 0 \right)$
or $\frac{q}{\mathcal{Q}} = \frac{1}{2}$
$\Longrightarrow$ $\ \ \ \ \ \ \ \ q = \frac{\mathcal{Q}}{2}$