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## Homework Statement

A constant voltage source V with internal resistance r is connected to a load resistor R. The dissipated power by the resistor R is P=RV^2/(R+r)^2. Show that the maximum power dissipated by the resistor R is achieved when R = r. The maximum of P with respect to R is achieved when dP/dR = 0.

## Homework Equations

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P = RV^2/(R+r)^2

## The Attempt at a Solution

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Well, I think that I need to find the absolute maximum. That means I need to find the first derivative, find the critical points, evaluate the equation at the critical points and find the absolute maximum. And somehow show that R = r.

To find the first derivative, I used the chain rule and came up with this:

f(g(x)) -> f'(g(x))g'(x)

P = RV^2/(R+r)^2 f = RV^2/x^2 g = R+r

f' = (2Vx - 2xRV^2)/x^4 g' = 0

P'= (2V(R+r) - 2(R+r)RV^2)/(R+r)^4

= (2V - 2RV^2)/(R+r)^3

(2V - 2RV^2)/(R+r)^3 = 0

I don't know if this is correct and I don't know how to go on from here. The denominator can't be zero because the derivative won't exist at those points. But if I set the nominator to zero, it doesn't really help me because there's no r, only R. What am I doing wrong?

All help is appreciated, thanks!