Question

Find the intervals of monotonicity for the following function.

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

• A. I in $(0,2)$ & D in $(-\infty ,0)\cup (2,\infty )$
• B. I in $(0,3)$ & D in $(-\infty ,0)\cup (3,\infty )$
• C. I in $(0,4)$ & D in $(-\infty ,0)\cup (4,\infty )$
• D. I in $(0,1)$ & D in $(-\infty ,0)\cup (1,\infty )$

Answer 1

Answer: (A)

I in $(0,2)$ & D in $(-\infty ,0)\cup (2,\infty )$

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

$f^{\prime }\left(x\right)>0$ implies

$-x^2.e^{-x}+2x.e^{-x}>0$

$e^{-x}(2x-x^2)>0$

Hence, $x(2-x)>0$

$x>0$ and $x<2$.

Hence, the function is increasing for $(0,2)$ and decreasing for the $x\in R-(0,2)$.