How to calculate maximum power when value of load resistance is unknown?
The power in the load resistor is:  $$P_L= I^2R_L$$

$$I=\frac{V}{R_{th}+R_L}.$$

Therefore:

$$ P_L=(\frac{V}{R_{th}+R_L})^2R_L.$$

To obtain the maximum power, differentiate and assign zero to obtain maximum:

$$\frac{d P_L}{dR_L}=0 \text{ therefore } R_L=R_{th}  ~~~~~~\text{(NB.)}$$

This gives $$P_{max} = \frac{V^2}{4\cdot R_{th}}.$$