A battery with EMF E and internal resistance r is connected across...

Power consumed in external resistance R is $P = \frac{E^2 R}{(R + r)^2}$.
To find maximum, differentiate with respect to R:
$\frac{dP}{dR} = E^2 \frac{(R + r)^2 - 2R(R + r)}{(R + r)^4} = E^2 \frac{r - R}{(R + r)^3}$.
$\frac{dP}{dR} = 0$ gives $R = r$.
This is the maximum power transfer theorem: maximum power is delivered when the external load resistance equals the source internal resistance.