Problem Statement
The marginal cost of producing $x$ units of a product is given by $MC = 3x^2 - 12x + 10$. The fixed cost is ₹$100$. Find:
(i) The total cost function $C(x)$.
(ii) The average cost function $AC(x)$.
(iii) The total cost of producing $5$ units.
Verified Solution & Marking Scheme
Integrate Marginal Cost to Find Total Cost
$$C(x) = \int MC \, dx = \int (3x^2 - 12x + 10) \, dx = x^3 - 6x^2 + 10x + k$$
Use Fixed Cost Condition to Find Constant k
When $x = 0$, fixed cost $C(0) = 100 \implies 0 - 0 + 0 + k = 100 \implies k = 100$.
Thus, total cost function is:
$$C(x) = x^3 - 6x^2 + 10x + 100$$
Find Average Cost and Cost for 5 Units
Average cost $AC(x) = \frac{C(x)}{x} = x^2 - 6x + 10 + \frac{100}{x}$.
For $x = 5$ units:
$$C(5) = 5^3 - 6(5^2) + 10(5) + 100 = 125 - 150 + 50 + 100 = ext{Rs } 125$$