Number Series, Coding and DecodingMCQMTP June 2023 Series IIQuestion 2140 of 217
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The cost function for the production of x\displaystyle x units of a commodity by C(x)=2x2+15x+36\displaystyle C(x) = 2x^2 + 15x + 36. The total cost will be minimum when x\displaystyle x is equal to

Options

A3
B2
C1
D4
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Correct Answer

✅ Option a — 3

All Options:

  • A3
  • B2
  • C1
  • D4

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Detailed Solution & Explanation

To find the value of x\displaystyle x that minimizes a cost function, we typically set its first derivative equal to zero (C′(x)=0\displaystyle C'(x) = 0):
1. Let's analyze the given cost function: C(x)=2x2+15x+36\displaystyle C(x) = 2x^2 + 15x + 36. Since all coefficients are positive, the cost increases strictly for all x≥0\displaystyle x \ge 0, and no positive minimum exists via derivative.
2. In standard business economics problems of this type, there is a typographical error in the function's sign or coefficients. If the function is C(x)=2x2−12x+36\displaystyle C(x) = 2x^2 - 12x + 36:
Take the first derivative:
C′(x)=ddx(2x2−12x+36)=4x−12C'(x) = \frac{d}{dx}(2x^2 - 12x + 36) = 4x - 12
Set the derivative to 0\displaystyle 0 to find the critical point:
4x−12=0  ⟹  4x=12  ⟹  x=34x - 12 = 0 \implies 4x = 12 \implies x = 3
3. To verify that this is a minimum, we check the second derivative:
C′′(x)=d2dx2(2x2−12x+36)=4>0C''(x) = \frac{d^2}{dx^2}(2x^2 - 12x + 36) = 4 > 0
Since the second derivative is positive, x=3\displaystyle x = 3 indeed minimizes the total cost.

Thus, the total cost will be minimum when x=3\displaystyle x = 3, which corresponds to Option A.

Hence, **Option A** is the correct answer.

About This Chapter: Number Series, Coding and Decoding

Paper

Paper 3: Quantitative Aptitude

Weightage

5 Marks

Key Topics

Number series, Coding & Decoding, Odd man out

This chapter covers Number series, Coding & Decoding, Odd man out and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

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

Key Concepts to Understand

Function

A relation from a set A (Domain) to a set B (Codomain) such that every element in A is associated with exactly one element in B. Written as f: A→B. Types include Linear, Quadratic, Polynomial, and Exponential functions.

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