Question

If $f(x) = e^x (x - 2)^2$ then

• A. $f$ is increasing in $(-8,0)$ and $(2,8)$ and decreasing in $(0, 2)$
• B. $f$ is increasing in $(-8,0)$ and decreasing in $(0,8)$
• C. $f$ is increasing in $(2,8)$ and decreasing in $(-8,0)$
• D. $f$ is increasing in $(0, 2)$ and decreasing in $(-8,0)$ and $(2,8)$

Answer 1

Answer: (A)

$f$ is increasing in $(-8,0)$ and $(2,8)$ and decreasing in $(0, 2)$

Answer 2

Given function is, $f(x)=e^{x}(x-2)^{2}$
$\Rightarrow f^{\prime}(x) =e^{x}(x-2)^{2}+2(x-2) \theta^{x}$
$=e^{x}(x-2)(x-2+2)=x(x-2) e^{x}$
Now, sign scheme of $f^{\prime}(x)$ is

So, $f$ is increasing in $(-\infty, 0)$ and $(2, \infty)$ and decreasing in $(0,2)$.