Let `f:RtoR` be given by
`f(x)={{:(x^(5)+5x^(4)+10x^(3)+3x+1,xlt 0),(x^(2)-x+1,0 le x lt1),((2//3)x^(3)-4x^(2)+7x-(8//3),1 le x lt3),((x-2)ln(x-2)-x+(10//3),x ge 3):}`
Then which of the following options is/are correct?

Options

A. f is onto
B. f' is not differentiable at x=1
C. f' has a local maximum at x=1
D. f is increasing on (`-infty,0)`

Answer

A, B, C

Explanation

`f(x)={{:(x^(5),5x^(4)+10x^(3)+3x+1,x lt 0),(x^(2)-x+1,,0 le x lt 1),((2)/(3)x^(3)-4x^(2)+7x-(8)/(9),,1 le x lt3),((x-2)I n(x-2)-x+(10)/(3),,x ge 3):}`
`f(x)={{:(5(x+1)^(4)-2,xlt0),(2x-1,0lexlt1),(2x^(2)-8x+7,1lexlt3),(In(x-2),xge3):}`
`x^(5)+5x^(4)+10x^(2)+3x+1` takes value between `-infty` to 1
also `(x-2)` IN (x-2)-x`+(10)/(3)` takes value between `(1)/(3)` to `infty`
so range of f(x) is R. so option (A) is correct
`f''(1^(-))=2` and `f''(1^(+))=-4`
so f'(x) is non-diff at x=1 so option (B) is correct f'(x) has focal maxima at x=1 so option (C) is correct