Question

Find the interval of increase and decrease of the following functions.
$f\left(x\right)=x^2{e}^{-x}$

Answer 1

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

First, $f^{\prime }\left(x\right)=2x{e}^{-x}-x^2{e}^{-x}$

$\Rightarrow e^{-x}(2x-x^2)=0$

$\Rightarrow \underset{Notpossible}{\underset{}{e^{-x}=0}}$ and $x(2-x)=0$

$x=0$ or $x=2$

Critical points : $x=0$ and $x=2$

The function monotone intervals are :-

$-\infty <x<0,0<x<2,2<x<\infty$

At $-\infty <x<0$, let $x=-1$

$f^{\prime }(-1)=-3e<0$, therefore decreasing.

At $0<x<2$, let $x=1$

$f^{\prime }\left(x\right)=\frac{1}{e}>0$, therefore incresing

At $2<x<\infty$, let $x=3$,

$f^{\prime }\left(3\right)=\frac{-3}{e^3}$, therefore decreasing.

Interval of decrease, $-\infty <x<0$ and $2<x<\infty$

Interval of increase, $0<x<2$.