The interval in which `y=x^2""e^(-x)` is increasing is

Options

A. `(-infty, infty)`
B. `(-2, 0)`
C. `(2, infty)`
D. `(0, 2)`

Answer

D

Explanation

`y=x^(2)e^(-x)`
` rArr (dy)/(dx) =- x^(2)e^(-x) + 2xe^(-x)`
` rArr =-xe^(-x)(x-2)`
For increasing function
`-xe^(-x)(x-2) gt 0`
` rArr x(x-2) lt 0 " "(.:' e^(-x) gt)`
` rArr x gt 0 and x- 2 lt 0 or x lt 0 and x - 2 gt 0`
` rArr x gt 0 and x lt 2 or x lt 0 and x gt 2 `
Second part is not possible.
`:. x gt 0 and x lt 2`
` rArr 0 lt x lt 2`
` :. f(x) ` is increasing in (0, 2).