1.

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

Answer»

`0ltklte`
`0ltklt(8)/(e^(2))`
`0ltklt(4)/(e^(2))`
none of these

Solution :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


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