Discuss the monotonocity of function `f(x)=2log|x-1|-x^2+2x+3.`

Answer

Increasing :`(-oo,0)cup(1,2)`
Decreasing : `(0,1)cup(2,oo)`

Explanation

`f(x) =2log |x-1| -x^(2)+2x+3`
`therefore f'(x) =(2)/(x-1)-2x+2 =1[(1-(x-1)^(2))/(x-1)]=(-2x(x-2))/(x-1)`
sign scheme for `(-2x(x-2))/(x-1)` is as follows
`f'(x) gt 0 if x in (-oo,0) or x in (1,2)`
Therefore ,f(x) is increasing in the interval `(-oo,0)cupo(1,2)`
and decreases if `x in(0,1)cup(2,oo)`