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GATE 2022 CE (CE2) – Question 36

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

$f(x)=x^3-6x^2+11x-6$ on $1\le x\le3$. I: f is zero at x=1,3. II: one local maximum exists. III: $f^{\prime\prime}>0$ throughout. IV: one local minimum exists. Choose the correct combination.

  1. Only statements I, II and III are correct.
  2. Only statements I, II and IV are correct.
  3. Only statements I and IV are correct.
  4. Only statements II and IV are correct.

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Correct answer: (B) Only statements I, II and IV are correct.

Explanation

Factor $f=(x-1)(x-2)(x-3)$, proving I. Stationary points solve $3x^2-12x+11=0$, giving $x=2\pm1/\sqrt3$, both within [1,3].

Since $f^{\prime\prime}=6x-12$, the smaller point is a maximum and the larger a minimum. The second derivative changes sign at 2, so III is false. **I,II,IV: B**.