Let `f(x) `be real valued continous funcion on R defined as `f(x)` =`x^(2)e^(-|x|)`
Number of points of inflection for `y = f(x)` is (a) 1 (b) 2 (c) 3 (d) 4

Options

A. y = f(x) has two points of maxima
B. y=f(X) has only one asymptote
C. f(X) =0 has three real roots
D. none of these

Answer

4

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