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GATE CS 2012 · General Aptitude

Question 56

MCQPrinted as GA Q.561 markWrong answer: −1/3Easy (estimate)

Question

The cost function for a product in a firm is given by 5q25q^2, where qq is the amount of production. The firm can sell the product at a market price of Rs 50 per unit. The number of units to be produced by the firm such that the profit is maximized is

Choose one option
Official answer and solution
Official answer
A
From
Official GATE 2012 answer key

Solution

Maximize the firm's profit function using differential calculus.

  1. Cost function: C(q)=5q2C(q) = 5q^2.
    Selling price per unit: P=50P = 50.
    Revenue function: R(q)=50qR(q) = 50q.

  2. Profit function:

Π(q)=R(q)−C(q)=50q−5q2\Pi(q) = R(q) - C(q) = 50q - 5q^2
  1. Setting marginal profit to zero:
dΠdq=50−10q=0  ⟹  10q=50  ⟹  q=5\frac{d\Pi}{dq} = 50 - 10q = 0 \implies 10q = 50 \implies q = 5
  1. Second derivative test:
d2Πdq2=−10<0\frac{d^2\Pi}{dq^2} = -10 < 0

Confirming q=5q = 5 is a profit-maximizing point.

  1. Thus, the firm should produce 5 units (Option A).
Final answer

A

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper
GATE 2012 Computer Science and Information Technology (CS)
Question
56 of 65, printed as GA Q.56
Section
General Aptitude

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