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GATE 2023 CS – Question 28

Engineering Mathematics · Calculus · 1 mark · Multiple select

Let $f(x) = x^3 + 15x^2 - 33x - 36$ be a real-valued function. Which of the following statements is/are TRUE?

  1. f(x) does not have a local maximum.
  2. f(x) has a local maximum.
  3. f(x) does not have a local minimum.
  4. f(x) has a local minimum.

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Show answer and explanation

Correct answer: (B) f(x) has a local maximum.; (D) f(x) has a local minimum.

Explanation

f'(x)=3(x+11)(x−1), so the critical points are x=−11 and x=1. f''(x)=6x+30 is negative at −11 (local max) and positive at 1 (local min).