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Applications of Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives2 Marks
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Answer

Correct option: D.
monotonically decreasing in $(-1, \infty)$ and monotonically increasing in $(-\infty,-1)$
(d) : Given, $f(x)=(x+2) e^{-x}$
$f^{\prime}(x)=-(x+2) e^{-x}+e^{-x}=-e^{-x}(x+1)$
For increasing function, $-e^{-x}(x+1)>0$
i.e., $e^{-x}(x+1)<0$
Since, $e^{-x}>0$, therefore $(x+1)<0$
$\Rightarrow x \in(-\infty,-1)$
$\therefore$ For $x \in(-\infty,-1)$, function is increasing.
Now, for decreasing function, $-e^{-x}(x+1)<0$
i.e. $e^{-x}(x+1)>0 \Rightarrow x \in(-1, \infty)$
$\therefore$ For $x \in(-1, \infty)$, function is decreasing.

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