Mean Value theorem problem? Im 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

likely App is electrical (or electron flow). I am just an economist BUT here goes.

for w = |e|^2 R/(R + Ri)^2 to be max where R = Ri you use the properties of the Mean Value thm, that basically allow you to say that there is a MAX when the first derivative = 0 (with some important stuff that go with it.

Proving that w maxes where R = Ri means that dw/dR = 0 where R = Ri

Figuring that is what is being asked is the hard part.

dw/dR = |e|² [ (R+R_i)² - 2R(R + R_i] / (R + R_i)⁴ and for a max that has to equal zero

lets expand |e|² [ R² + 2 R R_i + R_i² - 2R² - 2R R_i ] / (R + R_i) = 0 divide out |e|² and combine stuff

[-R² + R_i²] / (R + R_i) = 0 that means R² = R_i² or R = R_i

check second order stuff and you'll find w" < 0 which means you're good

I think that is what you are looking for.