Function y=x^2 e^ -x is decreasing in which of interval : - Vidyadip

MCQ

Function $y=x^2 e^{-x}$ is decreasing in which of interval :

A
$(0,2)$
B
$(2, \infty)$
C
$(-\infty, 0)$
✓
$(-\infty, 0) \cup(2, \infty)$

✓

Answer

Correct option: D.

$(-\infty, 0) \cup(2, \infty)$

(D) Given $\quad y=f(x)=x^2 e^{-x}$$
\begin{aligned}
\therefore \quad f^{\prime}(x) & =x^2 e^{-x}(-1)+2 x e^{-x} \\
& =x e^{-x}(2-x)
\end{aligned}
$
now $e^{-x}>0 \quad \forall x \in R$
Hence $f(x)$ will be decreasing if$
\begin{aligned}
f^{\prime}(x)<0 \\
\Rightarrow x(2-x)<0 \\
\Rightarrow x(x-2)>0 \\
\Rightarrow &x \in(-\infty, 0) \cup(2, \infty)
\end{aligned}
$

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