The battery has an internal resistance of $4 \mathrm{\Omega }$.

From maximum power transfer theorem, we know that for maximum power, the equivalent resistance of the network should be equal to the internal resistance of the battery.

${\mathrm{R}}_{\mathrm{eq}}=4 \mathrm{\Omega }$

$\Rightarrow \frac{\left(7\text{R}\right)\left(12\text{R}\right)}{19\text{R}}=\frac{84\text{R}}{19}=4 \mathrm{\Omega }$

$\Rightarrow \text{R}=\frac{19}{21} \mathrm{\Omega }$