Q.The total cost of producing $x$ units is $C(x) = x^2 + 40x + 400$ (in ₹). Find the output at which the average cost is minimum, and show that at this output the average cost equals the marginal cost.

Minimise the average cost $\overline{C} = C(x)/x$ by setting its derivative to zero; then compare with marginal cost.

Average cost.

Minimise. Differentiate with respect to $x$:

Set $\overline{C}'(x) = 0$:

Confirm a minimum.

so $x = 20$ gives a minimum average cost. Its value:

So the minimum average cost is ₹80 per unit.

Compare with marginal cost.

$\text{MC} = \frac{dC}{dx} = 2x + 40, \qquad \text{MC}(20) = 2(20) + 40 = 80.$ …

Rejecting the negative root of $x^2 = 400$ is essential — output $x$ must be positive, so $x = 20$, not $x = -20$. Also, minimise the average cost $C(x)/x$, not the total cost …