The relation between 'marginal cost (MC)' and 'average cost (AC)' of producing 'x' units of a product is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{d}{dx}(AC)=\frac{1}{x}(MC-AC) \)
Let C(x) be the total cost. Then \( AC=\frac{C(x)}{x} \) and \( MC=C'(x) \). By the quotient rule:
\( \displaystyle \frac{d}{dx}(AC)=\frac{x\,C'(x)-C(x)}{x^{2}} \)
\( \displaystyle =\frac{1}{x}\left(C'(x)-\frac{C(x)}{x}\right) \)
\( \displaystyle =\frac{1}{x}(MC-AC) \)
Tip: This relation explains a standard fact: AC falls while MC < AC, rises while MC > AC, and is minimum where MC = AC.