Basic Applications of CalculusPYQ Sept 25Question 4137 of 28
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The cost function of an organisation as C(x)=500−5x2+x33\displaystyle C(x) = 500-5x^2+\frac{x^3}{3}, where x\displaystyle x denotes the output. Find the level of output at which marginal cost is the minimum.

Options

A5
B4
C10
D6
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Correct Answer

✅ Option a — 5

All Options:

  • A5
  • B4
  • C10
  • D6

Detailed Solution & Explanation

Given cost function: C(x)=500−5x2+x33C(x) = 500 - 5x^2 + \frac{x^3}{3}
The Marginal Cost (MC\displaystyle MC) is the derivative of the cost function with respect to x\displaystyle x: MC=ddxC(x)=ddx(500−5x2+x33)MC = \frac{d}{dx} C(x) = \frac{d}{dx}\left(500 - 5x^2 + \frac{x^3}{3}\right) MC=−10x+x2=x2−10xMC = -10x + x^2 = x^2 - 10x
To find the level of output x\displaystyle x at which the marginal cost is minimum, we differentiate MC\displaystyle MC with respect to x\displaystyle x and set it to 0: ddx(MC)=2x−10\frac{d}{dx}(MC) = 2x - 10 Set this equal to 0: 2x−10=0  ⟹  2x=10  ⟹  x=52x - 10 = 0 \implies 2x = 10 \implies x = 5
To verify that this is a minimum, we find the second derivative: d2dx2(MC)=2>0\frac{d^2}{dx^2}(MC) = 2 > 0 Since the second derivative is positive, x=5\displaystyle x = 5 is the point of minimum marginal cost.
Hence, **Option A** is the correct answer.

About This Chapter: Basic Applications of Calculus

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Limits, Continuity, Derivatives, Integrals

This chapter covers Limits, Continuity, Derivatives, Integrals and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

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