Q.The total cost of producing $x$ units is $C(x) = 3x^2 + 12x + 75$ (in ₹). Find the output at which the average cost is minimum and the minimum average cost.

Minimise the average cost $\overline{C} = C(x)/x$ by setting its derivative to zero and confirming with the second derivative.

Average cost.

Minimise. Differentiate:

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

Confirm a minimum.

so $x = 5$ gives a minimum.

Minimum average cost.

$\overline{C}(5) = 3(5) + 12 + \frac{75}{5} = 15 + 12 + 15 = 42.$ …

Take only the positive root of $x^2 = 25$ (output cannot be negative). And divide every term of $C(x)$ by $x$ when forming the average — including th …