The marginal cost (MC) and marginal revenue (MR) functions of a product are \(\mathrm{MC} = 20 + \displaystyle\frac{x}{20}\) and \(\mathrm{MR} = 30\) respectively. If the fixed cost is 200, then the maximum value of the profit is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 800
The total cost is the integral of MC, with the constant equal to the fixed cost; the revenue is the integral of MR, with no revenue when nothing is sold:
\( \displaystyle C(x) = \int \left(20 + \frac{x}{20}\right) dx = 20x + \frac{x^2}{40} + 200 \)
\(R(x) = 30x\)
The profit is
\( \displaystyle P(x) = 30x - 20x - \frac{x^2}{40} - 200 \)
\( \displaystyle = 10x - \frac{x^2}{40} - 200 \)
Profit is greatest where \(P'(x) = 0\), that is, where MR = MC:
\( \displaystyle P'(x) = 10 - \frac{x}{20} = 0 \)
\(x = 200\)
\( \displaystyle P''(x) = -\frac{1}{20} \lt 0 \)
so x = 200 gives the maximum:
\( \displaystyle P(200) = 2000 - \frac{40000}{40} - 200 \)
\(= 2000 - 1000 - 200 = 800\)
So the maximum profit is Rs. 800.