The total cost (in Rs.) of producing x units of an item is \(C(x) = 2x^{2} + 50x + 800\). The least possible average cost per unit is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 130
The average cost per unit is the total cost divided by the number of units:
\( \displaystyle AC(x) = \frac{C(x)}{x} = 2x + 50 + \frac{800}{x} \)
Setting the derivative equal to zero:
\( \displaystyle AC'(x) = 2 - \frac{800}{x^{2}} = 0 \)
\( x^{2} = 400,\ x = 20 \)
Since \(AC''(x) = \frac{1600}{x^{3}} \gt 0\), this is the minimum:
\( AC(20) = 40 + 50 + 40 = 130 \)
Why the other options are incorrect
Option (1) Rs. 90 — Incorrect. This is 2x + 50 at x = 20; the fixed-cost share \(\frac{800}{20} = 40\) is left out.
Option (2) Rs. 20 — Incorrect. 20 is the number of units at which the average cost is least, not the cost.
Option (3) Rs. 150 — Incorrect. The average cost reaches Rs. 130 at x = 20, so 150 is not the least value.