Number Series, Coding, Odd Man OutMTP 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

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.

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