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GATE 2015 EC – Question 38

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Which one of the following graphs describes the function $f(x) = e^{-x}(x^2 + x + 1)$?

Four graphs of $f(x)$ against $x$. (A) falls steeply and then flattens. (B) rises slightly, peaks, and then decays smoothly. (C) falls to a minimum, rises to a small bump, then falls. (D) falls below zero and then oscillates.
  1. Graph (A)
  2. Graph (B)
  3. Graph (C)
  4. Graph (D)

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Show answer and explanation

Correct answer: (B) Graph (B)

Explanation

We have $f(0) = 1$ and $f'(x) = e^{-x}\left(-x^2 - x - 1 + 2x + 1\right) = e^{-x} x(1 - x)$. So $f$ increases slightly for $0 < x < 1$ and then decreases, and it is always positive and tends to 0 as $x \to \infty$. That is graph (B).