A firm sells x units of a product at Rs. (100 − 2x) per unit. The total cost of producing x units is Rs. (20x + 300). The number of units the firm should sell to earn the maximum profit is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 20
The revenue from x units is \(R(x) = x(100 - 2x) = 100x - 2x^{2}\). The profit is revenue minus cost:
\( P(x) = 100x - 2x^{2} - (20x + 300) \)
\( = 80x - 2x^{2} - 300 \)
Setting the derivative equal to zero:
\( P'(x) = 80 - 4x = 0 \)
\( x = 20 \)
Since \(P''(x) = -4 \lt 0\), the profit is maximum at x = 20 (the maximum profit is P(20) = Rs. 500).
Why the other options are incorrect
Option (2) 25 — Incorrect. This maximises the revenue (R′(x) = 100 − 4x = 0); the cost has been ignored.
Option (3) 40 — Incorrect. P(40) = 3200 − 3200 − 300 = −300, a loss.
Option (4) 10 — Incorrect. P(10) = 800 − 200 − 300 = 300, less than P(20) = 500.