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