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

$(A) \;(-\infty, \: \infty) \quad (B)\; (-2,0) \quad (C)\; (2, \infty) \quad (D)\; (0,2)$

Toolbox:
• A function $f(x)$ is said to be a strictly increasing function on $(a,b)$ if $x_1 < x_2\Rightarrow f(x_1) < f(x_2)$ for all $x_1,x_2\in (a,b)$
• If $x_1 < x_2\Rightarrow f(x_1) > f(x_2)$ for all $x_1,x_2\in (a,b)$ then $f(x)$ is said to be strictly decreasing on $(a,b)$
• A function $f(x)$ is said to be increasing on $[a,b]$ if it is increasing (decreasing) on $(a,b)$ and it is increasing (decreasing) at $x=a$ and $x=b$.
• The necessary sufficient condition for a differentiable function defined on $(a,b)$ to be strictly increasing on $(a,b)$ is that $f'(x) > 0$ for all $x\in (a,b)$
• The necessary sufficient condition for a differentiable function defined on $(a,b)$ to be strictly decreasing on $(a,b)$ is that $f'(x) < 0$ for all $x\in (a,b)$
Step 1:
Let $f(x)=x^2e^{-x}$
Differentiating w.r.t $x$ (by applying product rule)
$f'(x)=2e^{-x}-x^2e^{-x}$
$\qquad=xe^{-x}(2-x)$
$f'(x)=0$
$\Rightarrow xe^{-x}(2-x)=0$
$\Rightarrow x=0$ and $x=2$
The points $x=0$ and $x=2$,divide the real line into three disjoint intervals namely.
$(-\infty,0),(0,2)$ and $(2,\infty)$
Step 2:
Consider the intervals $(-\infty,0)$ and $(2,\infty)$
$f'(x) < 0$ as $e^{-x}$ is always positive.
Therefore $f'(x)$ is decreasing on $(-\infty,0)$ and $(2,\infty)$
Step 3:
Consider the interval $(0,2),f'(x) >0$
$f$ is strictly increasing on $(0,2)$
Hence the correct answer is $D$