Find the intervals in which f(x) = (x + 2)e^ -x is increasing or … - Vidyadip

Question

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

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Answer

$f(x) = (x + 2)e^{-x}$
$f'(x) = -e^{-x}(x + 2) + e^{-x}$
$= -xe^{-x} - 2e^{-x} + e^{-x}$
$= -xe^{-x} - e^{-x}$
$= e^{-x}(-x - 1)$
For f(x) to be increasing, we must have
$f'(x) > 0$
$\Rightarrow e^{-x}(-x - 1) > 0$
$⇒ -x - 1 > 0$ $\big[\because\ \text{e}^{-\text{x}}>0,\forall\ \text{x}\in\text{R}\big]$
$⇒ -x > 1$
$⇒ x < -1$
$\Rightarrow\text{x}\in(-\infty,-1)$
So, f(x) is increasing on $(-\infty,-1).$
For f(x) to be decreasing, we must have
$f'(x) < 0$
$\Rightarrow e^{-x}(-x - 1) < 0$
$⇒ -x - 1 < 0$ $\big[\because\ \text{e}^{-\text{x}}>0,\forall\ \text{x}\in\text{R}\big]$
$⇒ -x < 1$
$⇒ x < -1$
$\Rightarrow\text{x}\in(-1,\infty)$
So, f(x) is decreasing on $(-1,\infty).$

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