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GATE 2022 EE – Question 41

Engineering Mathematics · Calculus: Maxima and minima · 2 marks · Multiple choice

Let $f(x)=\int_0^xe^t(t-1)(t-2)\,dt$. Then $f(x)$ decreases in the interval

  1. $x\in(1,2)$
  2. $x\in(2,3)$
  3. $x\in(0,1)$
  4. $x\in(0.5,1)$

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Show answer and explanation

Correct answer: (A) $x\in(1,2)$

Explanation

$f'(x)=e^x(x-1)(x-2)$. Since $e^x>0$, $f'<0$ exactly when $(x-1)(x-2)<0$, i.e. for $1<x<2$.