GATE DA 2024 | Question 50

MSQ · 2 marks

Consider the function \(f: \mathbb{R} \to \mathbb{R}\) where \(\mathbb{R}\) is the set of all real numbers.

\[ f(x) = \frac{x^{4}}{4} - \frac{2x^{3}}{3} - \frac{3x^{2}}{2} + 1 \]

Which of the following statements is/are TRUE?

  1. Option A: \(x = 0\) is a local maximum of \(f\) (correct)
  2. Option B: \(x = 3\) is a local minimum of \(f\) (correct)
  3. Option C: \(x = -1\) is a local maximum of \(f\)
  4. Option D: \(x = 0\) is a local minimum of \(f\)

Answer

(A), (B)

Explanation

Differentiate and factor:

\[ f'(x) = x^{3} - 2x^{2} - 3x = x(x - 3)(x + 1) \]

so the critical points are \(x = -1, 0, 3\). Test each with the second derivative \(f''(x) = 3x^{2} - 4x - 3\):

  • \(f''(0) = -3 < 0\): a local maximum at \(x = 0\), so A holds and D fails.
  • \(f''(3) = 27 - 12 - 3 = 12 > 0\): a local minimum at \(x = 3\), so B holds.
  • \(f''(-1) = 3 + 4 - 3 = 4 > 0\): \(x = -1\) is a local minimum, not a maximum, so C fails.

Answer: A, B