Consider the function f: R R where R is the set of all real numbers. f(x) = x^4/4 - 2x^3/3 - 3x^2/2 + 1 Which of the following statements is/are TRUE?

GATE DA 2024 Q50: Local extrema of a quartic polynomial

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?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, B (official GATE 2024 final answer key)

Concept used: Stationary points and the second derivative test.

First step: $f'(x) = x^3 - 2x^2 - 3x = x(x-3)(x+1)$ and $f''(x) = 3x^2 - 4x - 3$ ; check the sign of $f''$ at $0$ , $3$ and $-1$ .

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key