Question

The function $f(x)=x^2{e}^{-x}$ increases in the interval

• A. $(0,2)$
• B. $(2,3)$
• C. $(3,4)$
• D. $(4,5)$

Answer 1

Answer: (A)

Given, $f(x)=x^2{e}^{-x}$
$\Rightarrow$ $f^{\prime }(x)=2x{e}^{-x}-x^2{e}^{-x}$
For $f(x)$ to be increasing, $f^{\prime }(x)>0$
$\Rightarrow$ $2x{e}^{-x}-x^2{e}^{-x}>0$
$\Rightarrow$ $e^{-x}(2x-x^2)>0$
$\Rightarrow$ $x(2-x)>0$
$\Rightarrow$ $x\in (0,2)$