The power dissipated in RI, PI=I2RI. The power dissipated in RO, PO=I2RO. The power generated by VO=IVO. The current is I=VO/(RI+RO). So,
The maximum (dPI/dRI=0) is at RI=RO and this is the maximum power.
Fischer-Cripps. A.C., The Electronics Companion. Institute of Physics, 2005.
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