The variable cost (V) of manufacturing a product varies according to the equation $V = 4q$ , where $q$ is the quantity produced. The fixed cost (F) of production of same product reduces with $q$ according to the equation $F = 100/q$ . How many units should be produced to minimize the total cost ( $V+F$ )?

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Solution

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Official answer: A) 5

Quantitative Aptitude

The total cost function is:

C(q) = V + F = 4q + \frac{100}{q}

To find the quantity $q$ that minimizes total cost, differentiate $C(q)$ with respect to $q$ and set the derivative to zero:

\frac{dC}{dq} = 4 - \frac{100}{q^2} = 0

4q^2 = 100 \implies q^2 = 25 \implies q = 5

Check the second derivative:

\frac{d^2C}{dq^2} = \frac{200}{q^3} > 0 \quad \text{for } q = 5

Thus, the total cost is minimized when $5$ units are produced, matching option (A).

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This Quantitative Aptitude previous year question is Q63 of the GATE CSE 2011 paper. Find more GATE CSE PYQs on Quantitative Aptitude in General Aptitude, or browse all GATE CSE PYQs.

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Quantitative Aptitude has 53 GATE CSE PYQs across 13 papers from 2011 to 2026. 27 of them are 2-mark MCQs like this one, and GATE CSE 2011 has 4.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.