Hi, I have struggled to understand the algebra and the reasoning, please elaborate !

It is not difcult to prove that when the marginal value is greater (less) than the average value, the average value is rising (falling). To prove this, we rst nd the maximum or min- imum of A, by taking the derivative with respect to x and setting it equal to zero. Since A E 31% is a quotient, we have to use the quotient rule of differentiation. This gives 91_ six =idz_dz_=0 (9.20) dx x2 Assuming x at 0, and since % = 1, this expression will equal zero when the numerator equals zero: that is, when dy x y = 0 which is true when dx '11:) dx x . But the left-hand side of this equation is the marginal value, M, and the right-hand side is the average value of the function, A. So the average value, A, reaches a maximum (or minimum) when M =A: that is, when the marginal and average values are equal. Using equation (9.20) with the eqqals sign replaced by > or 0) we must have M > A; and that when A is falling we dx must have M 1 if M> A, and EY