Question:
Find the intervals in which $f(x)=(x+2) e^{-x}$ is increasing or decreasing ?
Solution:
we have,
$f(x)=(x+2) e^{-x}$
$f^{\prime}(x)=e^{-x}-e^{-x}(x+2)$
$=e^{-x}(1-x-2)$
$=-e^{-x}(x+1)$
Critical points
$f^{\prime}(x)=0$
$\Rightarrow-e^{-x}(x+1)=0$
$\Rightarrow x=-1$
Clearly $\mathrm{f}^{\prime}(\mathrm{x})>0$ if $\mathrm{x}<-1$
$\mathrm{f}^{\prime}(\mathrm{x})<0$ if $\mathrm{x}>-1$
Hence $f(x)$ increases in $(-\infty,-1)$, decreases in $(-1, \infty)$
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