GATE 2024 DA Q.50: solution and answer

GATE 2024 · DAQ.50 · 2 marks · MSQ

Consider the function $f : R \to R$ where $R$ is the set of all real numbers.

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

Which of the following statements is/are TRUE?

(A)
$x = 0$ is a local maximum of $f$
(B)
$x = 3$ is a local minimum of $f$
(C)
$x = - 1$ is a local maximum of $f$
(D)
$x = 0$ is a local minimum of $f$

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Answer and solution

ANSWER

Options (A) and (B) are true: $x = 0$ is a local maximum and $x = 3$ is a local minimum of $f$ .

Official answer: A, BSource: Official answer key, GATE 2024 DA (IISc Bengaluru)

Differentiate: $f^{'} (x) = x^{3} - 2 x^{2} - 3 x = x (x^{2} - 2 x - 3) = x (x - 3) (x + 1)$ .
Stationary points (where $f^{'} (x) = 0$ ): $x = - 1, 0, 3$ .
Second derivative: $f^{''} (x) = 3 x^{2} - 4 x - 3$ .
At $x = 0$ : $f^{''} (0) = - 3 < 0$ , so $x = 0$ is a local maximum. Option (A) is true and option (D) is false.
At $x = 3$ : $f^{''} (3) = 27 - 12 - 3 = 12 > 0$ , so $x = 3$ is a local minimum. Option (B) is true.
At $x = - 1$ : $f^{''} (- 1) = 3 + 4 - 3 = 4 > 0$ , so $x = - 1$ is a local minimum, not a local maximum. Option (C) is false.
Cross-check with the sign of $f^{'} (x) = x (x - 3) (x + 1)$ : it is negative for $x < - 1$ (for example $f^{'} (- 2) = - 10$ ), positive on $(- 1, 0)$ (for example $f^{'} (- 0.5) = 0.875$ ), negative on $(0, 3)$ (for example $f^{'} (1) = - 4$ ) and positive for $x > 3$ . So $f$ falls then rises at $- 1$ (minimum), rises then falls at $0$ (maximum), and falls then rises at $3$ (minimum). This agrees with the second-derivative test.

KEY IDEA

Find the stationary points from $f^{'} (x) = 0$ , then classify each one with the second-derivative test (or the sign change of $f^{'}$ ): $f^{''} > 0$ gives a local minimum and $f^{''} < 0$ gives a local maximum.

WATCH OUT FOR

Mixing up the sign of $f^{''}$ . Here $f^{''} (- 1) = 4 > 0$ , so $x = - 1$ is a local minimum, and option (C) is a trap. Another slip is to treat $x = 0$ as a minimum, when $f^{''} (0) = - 3 < 0$ makes it a maximum.

From the GATE 2024 DA paper. Every question of this paper · Download the full paper · All 5 Maxima & minima questions, with notes

GATE 2024 DA Q.50: solution and answer

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