In the given circuit 'a' is an arbitrary constant. The value of $m$ for which the equivalent circuit resistance is minimum, will be $\sqrt{\frac{x}{2}}$. The value of $x$ is _____ .

Left part of the resistors are connected in parallel, hence $R_1=\frac{am}{3}$.

Similarly, for the right part of the resistors we can write, $R_2=\frac{a}{2m}$.

Therefore, equivalent resistance will be,

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

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

$\Rightarrow \frac{a}{3}=\frac{a}{2{\mathrm{m}}^2}$  $\Rightarrow m=\sqrt{\frac{3}{2}}$

Thus, $x=3$.