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GATE 2023 AE – Question 36

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 2 marks · Multiple choice

For $y=(x+3)(x-2)$ on $-4<x<4$, where is the minimum?

  1. $-3/2$
  2. $-1/2$
  3. $1/2$
  4. $3/2$

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Correct answer: (B) $-1/2$

Explanation

$$y=(x+3)(x-2)=x^2+x-6 .$$

**Stationary point.** $y^\prime=2x+1=0\Rightarrow x=-\dfrac12$.

**Nature.** $y^{\prime\prime}=2>0$, so this is a **minimum**. It lies inside the interval $-4<x<4$.

(The minimum value is $y(-\tfrac12)=\tfrac14-\tfrac12-6=-6.25$.)

Answer $x=-\tfrac12$ (B).