Application of Derivatives — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
Question
The interval in which $y = x^2 e^{–x}$ is increasing is
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Answer
It is given that $y = x^2 e^{–x}$
then $\frac{d y}{d x} = 2xe^{-x} - x^2e^{-x} = xe^{-x} (2-x)$
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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