Question

Investigate the behaviour of the following and construct their graph:

$y=x^2{e}^{-x}$

• A. Increasing for $(3,5)$ and decreasing for the $R-(3,5)$
• B. Increasing for $R-(0,2)$ and decreasing for the $(0,2)$
  
• C. Increasing for $(0,2)$ and decreasing for the $R-(0,2)$
  
• D. Increasing for $R-(3,5)$ and decreasing for the $(3,5)$
  

Answer 1

The correct option is C Increasing for $(0,2)$ and decreasing for the $R-(0,2)$

Hence $y=f\left(x\right)$ does not cut the $x$-axis at any point.
Now $f^{\prime }\left(x\right)>0$ implies $-x^2.e^{-x}+2x.e^{-x}>0$
$\Rightarrow e^{-x}(2x-x^2)>0$
Hence, $x(2-x)>0$
$x>0$ and $x<2$.
Hence, the function is increasing for $(0,2)$ and decreasing for the $xϵR-(0,2)$.