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

• Correct answer
  A

  $f$ is increasing in $(-\infty, 0)$ and $(2, \infty)$ and decreasing $\operatorname{lin}(0,2)$
• B

  $t$ is increasing in $(-\infty, 0)$ and decreasing in $(0, \infty)$
• C

  $f$ is increasing in $(2, \infty)$ and decreasing in $(-\infty, 0)$
• D

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

Solution

Given function is, $f(x)=e^{x}(x-2)^{2}$ $$ \begin{aligned} \Rightarrow f^{\prime}(x) &=e^{x}(x-2)^{2}+2(x-2) e^{x} \\ &=e^{x}(x-2)(x-2+2)=x(x-2) e^{x} \end{aligned} $$ Now, sign scheme of $f^{\prime}(x)$ is So, $t$ is increasing in $(-\infty, 0)$ and $(2, \infty)$ and decreasing in (0,2)

Hence, the correct answer is option: 'A'.