The total cost (in Rs.) of producing x units of an item is \(C(x) = 3x^2 - 36x + 150\). The minimum total cost is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Rs. 42
At the least cost the marginal cost \(C'(x)\) is zero:
\(C'(x) = 6x - 36 = 0 \Rightarrow x = 6\)
\(C''(x) = 6 \gt 0\)
By the second derivative test, C is minimum at x = 6:
\(C(6) = 3(36) - 36(6) + 150\)
\(C(6) = 108 - 216 + 150 = 42\)
So the minimum total cost is Rs. 42.
Why the other options are incorrect
Option (2) Rs. 150 — Incorrect. This is C(0), the fixed cost; the cost falls below it for outputs between 0 and 12.
Option (3) Rs. 258 — Incorrect. The sign of the term 36x has been dropped: 108 + 150 = 258.
Option (4) Rs. 90 — Incorrect. This is C(2) = 12 − 72 + 150; the cost is still falling at x = 2.