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GATE 2024 DA – Question 50

Calculus and Optimization · Taylor series, maxima and minima, single-variable optimization · 2 marks · Multiple select

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. $x = 0$ is a local maximum of $f$
  2. $x = 3$ is a local minimum of $f$
  3. $x = -1$ is a local maximum of $f$
  4. $x = 0$ is a local minimum of $f$

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Show answer and explanation

Correct answer: (A) $x = 0$ is a local maximum of $f$; (B) $x = 3$ is a local minimum of $f$

Explanation

The derivative is $f'(x) = x^3 - 2x^2 - 3x = x(x - 3)(x + 1)$, so the critical points are $-1$, $0$ and $3$. The second derivative is $f''(x) = 3x^2 - 4x - 3$. At $x = -1$ it is $4 > 0$ (local minimum), at $x = 0$ it is $-3 < 0$ (local maximum) and at $x = 3$ it is $12 > 0$ (local minimum). So A and B are true.