Robotics and Calculus?! Oh nos!

[Al Skierkiewicz]

Super,
What this problem addresses is the maximum terminal voltage of the battery in real world conditions. For our robot systems the voltage available at the terminals of the battery is affected by the voltage drop across the internal resistance, R in your problem. Although you are calculating for output power maximum we are more concerned with terminal voltage since a fall below 8 volts will reset the controller. As the battery discharges the internal resistance rises. If the current draw remains the same then the output voltage must fall due to the drop across the internal resistance. Increase the load current and the voltage drop internally also rises making less voltage available for the robot. The result is RC reset and 2-5 sec of lost match time.
As a side note, there are other resistances in the overall system. Mostly those are resistance in wire and connectors. The bottom line is to power your RC from a point as close to the battery as possible, and not at the end of a string of high current devices.