The value of the resistance R in figure is adjusted such that power dissipated in the $2\mathrm{\Omega }$ resistor is maximum. Under this condition

Current in the circuit is

$i=\frac{V}{R_{eff}}$

$R_{eff}= 2+\frac{8R}{8+R}$

Power dissipated in 2ohm resistor is

$P = i^{2}r=\left(\frac{V^2}{R_{eff}}\right)r=\frac{288}{{\left(2+{\displaystyle \frac{8R}{8+R}}\right)}^2}$

P is maximum when the denominator is minimum. 

Denominator is minimum when R is 0.

Power in 2 ohms resistor is 

$P =\frac{288}{{\left(2+{\displaystyle \frac{8\times 0}{8+0}}\right)}^2}=72watts$