The cost function for a product in a firm is given by 5q2, where q is the amount of production. The firm can sell the product at a market price of ₹ 50 per unit. The number of units to be produced by the firm such that the profit is maximized is
GATE ME - 2012,
Answer: A
Profit is maximised at 5 units.
Revenue is 50q and cost is 5q2, so:
\[ \begin{aligned} \pi(q) &= 50q-5q^{2} \\ \frac{d\pi}{dq} &= 50-10q = 0 \Rightarrow q = 5 \end{aligned} \]
The second derivative is -10, so this is a maximum, with profit 250 - 125 = 125.
At 10 units revenue equals cost (500 = 500) and profit is zero: that is the break-even point, not the optimum. 15 and 25 units make a loss.
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