Assertion (A): For $C(x) = x^{2} + 2x + 400$, the average cost and the marginal cost are equal at $x = 20$.

Reason (R): Average cost is least at the output where it equals marginal cost.

Explanation

1. 01

   $AC(20) = 20 + 2 + 20 = 42$ and $MC(20) = 2 \times 20 + 2 = 42$. A is true.

2. 02

   R is true: $\frac{d}{dx}\left(\frac{C}{x}\right) = \frac{xC' - C}{x^{2}} = 0$ exactly when $C' = \frac{C}{x}$. And $x = 20$ is that least point, so R explains A.