Let f(X) be real valued continous funcion on R defined as f(X) =`x^(2)e^(-|x|)`
The values of k for which the equation `x^(2)e^(-|x|)`=k has four real roots are

Options

A. `0ltklte`
B. `0ltklt(8)/(e^(2))`
C. `0ltklt(4)/(e^(2))`
D. none of these

Answer

3

Explanation

We have `f(X)=x^(2)e^(-|x|={{:(x^(2)e^(-x),xge0),(x^(2)e^(x),xlt0):}`
`therefore f(X) ={{:(e^(-x)(2x-x^(2)),xge0),(e^(x)(x^(2)+2x),xlt0):}`
F(x) increasing in `(-oo,-2)cup(0,2)` and f(x) decreasing `(-2,0)cup(2,o)` thus
`f(x)={{:(e^(-x)(x^(2)-4x+2),xge0),(e^(x)(x^(2)+4x+2y),xlt0):}`

f(x)=0 has four roots hence there are four points of inflection