Finding power absorbed by a voltage source
IB = (4V-E)/1.5Ω=(8V-2E)/3Ω

VB=E+IB*0.5Ω=E+(4V-E)/3=(2E+4V)/3

PB=IB*VB=(8V-2E)*(2E+4V)/9Ω &lt;=4W 

and it gets to the maximun when (8V-2E)=(2E+4V) (according to AM–GM inequality) => E=1V

Or, look (8V-2E)*(2E+4V) as a quadratic function of E and let it equal to zero to get the two roots, 4V and -2V. The mid point of these two roots, 1V, corresponds to the max point.