Describe, in your own words and in just a couple of sentences, what the max power theorem says. I have a reason for wanting to use that as the starting point, but we'll just have to see if I can make it pan out.

 

The power transferred from a supply source to a load is at its maximum when the resistance of the load is equal to the internal resistance of the source.

 

Let's clarify that just a bit. Saying, the "internal resistance of the source" implies a bit more than is really meant. A better way of putting it is that the power delivered to a load is maximized when the load resistance is set equal to the resistance of the circuit driving it, as seen by the load.

It's a bit of a nitpick, but if I gave you a circuit that had a 12V source with a 3Ω internal resistance and then connected that source to the series connection of a 9Ω resistor and a load resistor and then asked you to tell me what value of the load resistor will result in the maximum amount of power being delivered to the load (assuming you can't change anything else), then many people would say 3Ω because that's the internal resistance of the source. But the load neither knows nor cares about the distinction between resistances in the circuit that are internal to the source or not. What it knows is that, looking back into the circuit, it sees 12Ω and, therefore, a load of 12Ω is what will result in the max power being delivered to the load.

An equivalent way to view it is that the "source" whose "internal resistance" we are matching is the entire rest of the circuit when it is reduced to its Thevenin equivalent.

So, with that in mind, what is the resistance of the rest of the circuit in this problem as seen by the load?