Find the intervals in which

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 $(-1, \infty)$