MHT CET · Maths · Application of Derivatives
The function f defined by \(\mathrm{f}(x)=(x+2) \mathrm{e}^{-x}\) is
- A decreasing for all \(x \in \mathbb{R}\)
- B decreasing in \((-\infty,-1)\) and increasing in \((-1, \infty)\)
- C decreasing in \((-1, \infty)\) and increasing in \((-\infty,-1)\)
- D increasing for all \(x \in \mathbb{R}\)
Answer & Solution
Correct Answer
(C) decreasing in \((-1, \infty)\) and increasing in \((-\infty,-1)\)
Step-by-step Solution
Detailed explanation
\(f'(x) = \frac{\mathrm{d}}{\mathrm{d}x}((x+2)\mathrm{e}^{-x}) = \mathrm{e}^{-x} - (x+2)\mathrm{e}^{-x} = \mathrm{e}^{-x}(1 - x - 2) = -(x+1)\mathrm{e}^{-x}\) \(-(x+1)\mathrm{e}^{-x} = 0 \Rightarrow x+1 = 0 \Rightarrow x = -1\)
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