Consider the function f(x) = x^33 + 72x^2 + 10x + 1332, x ∈ [-8, 0]. Which of the following statements is/are correct?

The minimum value of the derivative of f is attained at x = -72

The maximum value of f is attained at x = -5

The minimum value of f is attained at x = -2

The minimum value of the derivative of f is attained at x = -72

GATE DA 2025 Q49: Maxima and minima of a cubic on [−8, 0]

Calculus and Optimization
2 marks
MSQ
No negative marking

Consider the function $f(x) = \dfrac{x^3}{3} + \dfrac{7}{2}x^2 + 10x + \dfrac{133}{2}$ , $x \in [-8, 0]$ . Which of the following statements is/are correct?

Try it: multiple select: one or more correct options

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

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Answer: C, D (official GATE 2025 final answer key)

Concept used: Global extrema on a closed interval: compare critical points with the endpoints.

First step: Find the critical points from $f'(x) = x^2 + 7x + 10 = (x + 2)(x + 5)$ , then evaluate $f$ at $x = -8, -5, -2$ and $0$ before deciding the maximum and minimum.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE DA 2025 Q49: Maxima and minima of a cubic on [−8, 0]

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