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GATE 2025 CE (CE2) – Question 44

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 2 marks · Multiple select

Consider the function given below and pick one or more CORRECT statement(s) from the following choices.
$f(x) = x^3 - \frac{15}{2}x^2 + 18x + 20$

  1. f(x) has a local minimum at x = 3.
  2. f(x) has a local maximum at x = 3.
  3. f(x) has a local minimum at x = 2.
  4. f(x) has a local maximum at x = 2.

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Show answer and explanation

Correct answer: (A) f(x) has a local minimum at x = 3.; (D) f(x) has a local maximum at x = 2.

Explanation

The derivative is $f'(x) = 3x^2 - 15x + 18 = 3(x - 2)(x - 3)$, so the stationary points are $x = 2$ and $x = 3$. The second derivative is $f''(x) = 6x - 15$. At $x = 2$, $f'' = -3 < 0$ (local maximum), and at $x = 3$, $f'' = 3 > 0$ (local minimum).