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NDA 2020 Maths · Mathematics · Application of Derivatives · medium

The function f(x)=xe^-x is increasing in the interval

Single correct option

The function f(x)=xe−xf(x)=xe^{-x} is increasing in the interval
  1. A. (1,∞)(1,\infty)
  2. B. (0,∞)(0,\infty)
  3. C. (−1,1)(-1,1)
  4. D. (−∞,1)(-\infty,1)
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Answer: (D) (−∞,1)(-\infty,1)

f′(x)=e−x−xe−x=e−x(1−x)f'(x)=e^{-x}-xe^{-x}=e^{-x}(1-x). Since e−x>0e^{-x}>0, the sign of f′f' is the sign of 1−x1-x, which is positive for x<1x<1. So ff increases on (−∞,1)(-\infty,1) and decreases on (1,∞)(1,\infty), with a maximum value 1e\dfrac1e at x=1x=1. The interval (−1,1)(-1,1) is only a subset of the increasing region, so it is not the full answer, and (0,∞)(0,\infty) includes the decreasing part.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake