`f(x)=|x^2-3|x|+2|dot` Then which of the following is/are true `f^(prime)(x)=2x-3` for `x` in `(0,1)uu(2,oo)`
`f^(prime)(x)=2x+3` for `x` in `(-oo,-2)uu(-1,0)`
`f^(prime)(x)=-2x-3` for `x` in (-2,-1)`
None of these

Options

A. `f'(x)=2x-3" for " x in (0,1) cup (2, oo)`
B. `f'(x)=2x +3 for x in (-oo,-2) cup (-1,0)`
C. `f'(x)=-2x-3" for "x in (-2,-1)`
D. None of these

Explanation

`f(x)=|x^(2)-3x+2|, xge0`
`={{:(|x^(2)-3x+2|",",xge0),(|x^(2)+3x+2|",",xlt0):}`
`{{:(x^(2)-3x+2,x^(2)-3x+2ge,xge0),(-x^(2)+3x-2, x^(2)-3x+2lt0,xge0),(x^(2)+3x+2,x^(2)+3x+2ge0,xlt0),(-x^(2)-3x-2,x^(2)+3x+2lt0,xlt0):}`
`={{:(x^(2)-3x+2",",x in [0, 1] cup[2, oo]),(-x^(2)+3x-2",",x in (1,2)),(x^(2)+3x+2",", x in (-oo,-2]cup[-1,0]),(-x^(2)-3x+2",", x in (-2,-1)):}`
`"or "f'(x)={{:(2x-3",", x in (0,1) cup (2,oo)),(-2x+3",", x in (1,2)),(2x+3",", x in (-oo,-2)cup(-1,0)),(-2x-3",", x in (-2,-1)):}`