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Official PYQ · 2011Q63MCQ1 mark

GATE CS 2011 Q63 — General Aptitude

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

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Answer: A

The total cost C(q)C(q) is the sum of the variable cost VV and the fixed cost FF:

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

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

dCdq=4−100q2=0\frac{dC}{dq} = 4 - \frac{100}{q^2} = 0

Solving for qq:

4=100q24 = \frac{100}{q^2}q2=25q^2 = 25q=5(since q>0)q = 5 \quad (\text{since } q > 0)

To confirm that this value of qq minimizes the cost, we check the second derivative:

d2Cdq2=200q3\frac{d^2C}{dq^2} = \frac{200}{q^3}

For q=5q = 5, the second derivative is 200125=1.6>0\frac{200}{125} = 1.6 > 0, which confirms a local minimum.

Thus, 5 units should be produced to minimize the total cost.

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