Question

The function $f\left(x\right)$ is defined by $f\left(x\right)=(x+2)e^{-x}$is :

• A. decreasing for all $x$
• B. decreasing in $(-\infty ,-1)$ and increasing in $(-1,\infty )$
• C. increasing for all $x$
• D. decreasing in $(-1,\infty )$ and increasing in $(-\infty ,-1)$

Answer 1

The correct option is D

decreasing in $(-1,\infty )$ and increasing in $(-\infty ,-1)$

Explanation for correct option

Step 1: Find the derivative of $f\left(x\right)$

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

So,

$$\begin{gathered} f\text{'}\left(x\right)=(x+2)e^{-x} \\ =e^{-x}-(x+2)e^{-x} \\ =e^{-x}(1-x-2) \\ =-(x+1)e^{-x} \end{gathered}$$

Step 2: Find the critical points

The critical point of $f\left(x\right)$ are the values of $x$ for which $f\text{'}\left(x\right)=0$.

So, the critical points are:
$f\text{'}\left(x\right)=0$

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

$\Rightarrow x+1=0$

$\Rightarrow x=-1$

So, the number line showing the points of increasing and decreasing is drawn as:

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

Hence, option(D) is correct .