In the given circuit 'a' is an arbitrary constant. The value of m...

Left part of the resistors are connected in parallel, hence ${R}_{1} = \dfrac{am}{3}$.

Similarly, for the right part of the resistors we can write, ${R}_{2} = \dfrac{a}{2m}$.

Therefore, equivalent resistance will be,

$R = \dfrac{am}{3} + \dfrac{a}{2m}$

Differentiating both sides w.r.t $m$ to observe critical points, we get $\dfrac{dR}{dm} = \dfrac{a}{3} + \dfrac{a\left(- 1\right)}{2{m}^{2}} = 0$

$⇒ \dfrac{a}{3} = \dfrac{a}{2{m}^{2}}$ $⇒ m = \sqrt{\dfrac{3}{2}}$

Thus, $x = 3$.