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GATE 2024 CE (CE2) – Question 14

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 1 mark · Multiple choice

The function $f(x) = x^3 - 27x + 4$, $1 \le x \le 6$ has

  1. Maxima point
  2. Minima point
  3. Saddle point
  4. Inflection point

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Correct answer: (B) Minima point

Explanation

The derivative $f'(x) = 3x^2 - 27 = 0$ gives $x = \pm 3$, and only $x = 3$ is in the interval. The second derivative is $f''(3) = 18 > 0$, so $x = 3$ is a minimum. (The endpoints give higher values, so there is no interior maximum.)