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GATE 2022 CH – Question 14

Engineering Mathematics · Calculus: Definite and improper integrals, partial and total derivatives, maxima and minima · 1 mark · Multiple choice

The value of the real variable $x \ge 0$, which maximizes the function $f(x) = x^e e^{-x}$ is

  1. $e$
  2. 0
  3. $1/e$
  4. 1

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Show answer and explanation

Correct answer: (A) $e$

Explanation

$\ln f = e\ln x - x$, and its derivative $\frac{e}{x} - 1 = 0$ gives $x = e$. The second derivative is negative, so this is the maximum.