Home › ISC Class 12 › Mathematics › Application of Derivatives › The interval in which $y=x^2 e^{-x}$ is increasi…
The interval in which $y=x^2 e^{-x}$ is increasing is
A$(-\infty,\infty)$
B$(-2,0)$
C$(2,\infty)$
D$(0,2)$
Answer & Solution
Correct answer: D. $(0,2)$
1. $y'=2x e^{-x}-x^2 e^{-x}=e^{-x}x(2-x)$.
2. Since $e^{-x}>0$, sign of $y'$ follows $x(2-x)$.
3. $x(2-x)>0$ when $0<x<2$, so $y$ is increasing on $(0,2)$.
4. Outside $[0,2]$ the product is negative (decreasing), ruling out the other intervals.
_Source: NCERT Class 12 Mathematics Ch 6 "Application of Derivatives", p.12_
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