Mean Value theorem question? I'm actually not sure what the topic is, but PLEASE HELP

Question

We quote from V. Belevitch, Classical Network Theory (Holden-Day, 1968, p.159): “When a generator of e.m.f. e and internal positive resistance Ri is connected to a positive load resistance R, the active power dissipated in the load... w = |e|^2 R/(R + Ri)^2... is maximum with respect to R for R = Ri , that is, when the load resistance is matched to the internal resistance of the generator.” Verify this statement.

Answer 1

This is a problem involving a series circuit with one EMF and, in series with the EMF, an internal resistance Ri and a load resistance, which I will denote simply with R. The current flowing through the entire circuit, and hence also through the load resistance, as a function of R, is i= EMF/(Ri + R), and the power dissipated by R, as a function of R, is P(R) = i*i*R

=(EMF^2*R)/(Ri+R)^2.

In order to find the maximum (extremum) of P(R), we differentiate P(R) with respect to R and set to zero.

This yields an extremum at R=Ri.

In order to show that Ri is a max (and not a min or an inflection point), I just graphed P(x) =x/(3+x)^2 and saw a max at x=3. This suggests that P(x) = x/(n+x)^2 has a max at n, and therefore that P(R) has a max at Ri.

This verifies the statement that was made.