The function `f(x)=2log(x-2)-x^2+4x+1` increases on the interval (a) `(1,\ 2)` (b) `(2,\ 3)` (c) `(1,\ 3)` (d) `(2,\ 4)`

Options

A. (1,2)
B. (2,3)
C. (1,3)
D. (2,4)

Answer

B

Explanation

Clearly f(x) is defined for all ` x gt 2`
We have
`f(x)=2 log (x-2)-x^2 +4x +1 `
`rArr f(x) =2/(x-2)-2x+4`
`rArr f(x)=2/(x-2)-2x+4`
`rArr f(x) =2 {(1-(x-2)^2)/(x-2)}=(2(x-1)(x-3))/(x-2)`
` therefore f'(x) gt 0`
`rArr -2(x-1)(x-3)(x-2) gt 0`
`rArr -2(x-1)(x-3)(x-2) gt 0`
`rArr -2(x-1)(x-3) lt 0 rArr x in (-oo,1) cup (2,3)`

Clearly domain of f(x) is `(2,oo)` Thus f(x) is increasing on (2,3)