The function $\mathrm{f}(\mathrm{x})$ is defined by $\mathrm{f}(\mathrm{x})=(\mathrm{x}+2)…

1. monotonically decreasing in $(-1, \infty)$ and monotonically increasing in $(-\infty,-1)$
2. decreasing for all $\mathrm{x}$
3. increasing for all $\mathrm{x}$
4. decreasing in $(-\infty,-1)$ and increasing in $(-1, \infty)$

Solution

$\begin{aligned} & \mathrm{f}(\mathrm{x})=(\mathrm{x}+2) \mathrm{e}^{-\mathrm{x}} \Rightarrow \mathrm{f} \mathrm{P}^{\prime}(\mathrm{x})=\mathrm{e}^{-\mathrm{x}}-(\mathrm{x}+2) \mathrm{e}^{-\mathrm{x}}=\mathrm{e}^{-\mathrm{x}}(1-\mathrm{x}-2) \\ & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{e}^{-\mathrm{x}}(1+\mathrm{x}) \text { sign scheme }\end{aligned}$

Asked in: MHT CET 2022 (05 Aug Shift 2)

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