Power delivered to load $RL= P = I2RL = \dfrac{V^2RL}{(RS+RL)^2}$ to determine the value of RL for the maximum power to be transferred to the load,

$\dfrac{dP}{dRL} = 0 \\ \dfrac{dP}{dRL}= \dfrac{d}{dRL} V^2RL / (RS+RL)^2 \\ = V^2 [(RS+RL)^2 – (2RL) (RS+RL)] \\ (RS+RL)^4 \\ (RS+RL)^2 – 2 RL (RS+RL)^2 = 0 \\ RL^2 + RL^2 + 2RS RL - 2RS RL – 2RL^2 = 0 \\ RL^2 = R^2S \\ RL= RS$

Hence, the maximum power will be transferred to the load when load resistance is equal to the source resistance.