Application of Derivatives — MATHS STD 12 Science — Question
Rajasthan 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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