Maximum power of a two terminal device

The input impedance of a two terminal device may be show in
Picture below. The internal resistance Ri were connected in series
With the internal voltage source E. The connecting terminals as A
And B, and the open circuit voltage presented at these terminals is
E

if an external load R in connected to the device , the voltage
Presented at the output terminals will be dependent on the value of
R. The potential presented at the output terminals is

Eab = E*R/(R + Ri)

The power is given by

P = Eab/R =[ [E*R/(R + Ri)]^2] / R

And the maximizing condition

dP/dR = 0

is applied. There results

R = Ri