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GATE 2025 ME – Question 36

Engineering Mathematics · Calculus: Partial derivatives, Taylor series, maxima and minima, Fourier series · 2 marks · Multiple choice

In the closed interval $[0, 3]$, the minimum value of the function $f$ given below is

$$f(x) = 2x^3 - 9x^2 + 12x$$

  1. 0
  2. 4
  3. 5
  4. 9

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Correct answer: (A) 0

Explanation

The derivative is $f'(x) = 6x^2 - 18x + 12 = 6(x - 1)(x - 2)$, so the critical points are $x = 1$ and $x = 2$. The values are $f(0) = 0$, $f(1) = 5$, $f(2) = 4$ and $f(3) = 9$. The minimum value on $[0, 3]$ is 0, at $x = 0$.