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NDA 2026
Exam
NDA
Year
2026
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2026 · Mathematics · Application of Derivatives

Passage: Let S and T be the sets where...

Passage: Let SS and TT be the sets where f(x)=x33−5x22+6x+7f(x)=\dfrac{x^3}{3}-\dfrac{5x^2}{2}+6x+7 decreases and increases respectively.

Question: What is SS equal to ?

Options

  1. A{x≤2}∪{x≥3}\{x\leq2\}\cup\{x\geq3\}
  2. B{x<2}∪{x>3}\{x< 2\}\cup\{x > 3\}
  3. C(2,3)(2,3)
  4. D[2,3][2,3]

Correct answer

D. [2,3][2,3]

Step-by-step solution

Given f(x)=x33−5x22+6x+7f(x) = \dfrac{x^3}{3} - \dfrac{5x^2}{2} + 6x + 7

Differentiating with respect to xx, we get:

f′(x)=x2−5x+6f'(x) = x^2 - 5x + 6

f′(x)=(x−2)(x−3)f'(x) = (x - 2)(x - 3)

For the function to be decreasing, f′(x)≤0f'(x) \leq 0.

(x−2)(x−3)≤0(x - 2)(x - 3) \leq 0

x∈[2,3]x \in [2,3]

Therefore, the set SS where f(x)f(x) decreases is [2,3][2,3].

Answer: [2,3][2,3]
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