The interval in which y = x2 e–x is increasing is - Vidyadip

Question

The interval in which y = x² e^–x is increasing is

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Answer

It is given that y = x² e^–x
then $\frac{d y}{d x}$ = 2xe^-x - x²e^-x = xe^-x (2-x)
Now, if $\frac{d y}{d x}$ = 0
$\Rightarrow$ x = 0 and x =2
The points x = 0 and x= 2 divide the real line into three disjoint intervals ie, (-$\infty$,0), (0,2) and (2,$\infty$).
In interval (-$\infty$,0) and (2,$\infty$),
f’(x) < 0 as e^-x is always positive.
Therefore, f is decreasing on (-$\infty$,0) and (2,$\infty$).
In interval (0,2), f’(x) > 0
Therefore, f is strictly increasing in interval (0.2).

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