Question

Find the intervals in which $f\left(x\right)=(x+2)e^{-x}$ is increasing or decreasing.

Answer 1

Given: $f\left(x\right)=(x+2)e^{-x}$

To find: Intervals at which $f\left(x\right)$ is increasing on decreasing

Solution:

$f\left(x\right)=(x+2)e^{-x}$

$f^{\prime }\left(x\right)=\left(1\right)e^{-x}+(x+2)(-e^{-x})$

$=e^{-x}[1-x-2]$

$=e^{-x}[-x-1]$

$=-e^{-x}(1+x)$

Put $f^{\prime }\left(x\right)=0$

$\Rightarrow -e^{-x}(1+x)=0$

$e^{-x}=0$

$1+x=0$

$x=-1$

$f^{\prime }\left(x\right)>0$ $\forall x\in (-\infty ,-1)$

$\therefore f\left(x\right)$ increasing on $(-\infty ,-1)$

$f^{\prime }\left(x\right)<0$ $\forall x\in (-1,\infty )$

$\therefore f\left(x\right)$ is decreasing on $(-1,\infty )$.