The charge flowing in a conductor changes with time as Q(t) = - β t...

Given here, $Q = \left(\alphat - \beta{t}^{2} + \gamma{t}^{3}\right)$

Current flowing through conductor is given by

$i = \dfrac{d Q}{d t} = \left(\alpha - 2\betat + 3\gamma{t}^{2}\right) . . . \left(1\right)$

For current to be minimum, $\dfrac{d i}{d t} = 0$

So, $\dfrac{d i}{d t} = \left(6\gammat - 2\beta\right) = 0$

$⇒ t = \dfrac{\beta}{3\gamma}$

Substituting in equation (1),

$i = \left(\alpha - 2\betat + 3\gamma{t}^{2}\right) = \alpha - 2\beta\left(\dfrac{\beta}{3\gamma}\right) + 3\gamma{\left(\dfrac{\beta}{3\gamma}\right)}^{2} = \alpha - 2\dfrac{{\beta}^{2}}{3\gamma} + 3\dfrac{{\beta}^{2}}{9\gamma} = \left(\alpha - \dfrac{{\beta}^{2}}{3\gamma}\right)$