Question

The interval in which $y=x^2{e}^{-x}$ is increasing is

• A. $(-\infty ,\infty )$
• B. $(-2,0)$
• C. $(2,\infty )$
• D. $(0,2)$

Answer 1

The correct option is D $(0,2)$

The given function is $y=x^2{e}^{-x}$

It will strictly increase in those intervals where $\frac{dy}{dx}>0$

Now,

$\frac{dy}{dx}=-x^2{e}^{-x}+e^{-x}\left(2x\right)$

$=e^{-x}(2x-x^2)$

$e^{-x}$ will be always positive, so the function will strictly increase in those intervals where $2x-x^2>0$

The number line showing the sign of $2x-x^2$ is shown in the figure.

so the function will strictly increase in $(0,2)$