The interval in which the function
`f(x)=int_(0)^(x) ((t)/(t+2)-1/t)dt`will be non- increasing is

Options

A. `(-2,-1] cup (0,3]`
B. `(-2,-1 ] cup [0,3]`
C. `(-2,-1]cup [0,2]
D. `(-2,-1]cup ( 0,2]`

Answer

D

Explanation

We have `f(x)=underset(0)overset(x)int(t/(t+2)-1/t)dt`
`rArr f'(x)=(x)/(x+2)-1/x=(x^2-x-2)/(x(x+2))=((x-2)(x+1))/(x(x+2))`
For f(x) to be non-increasing , we must have
`f(x) lt 0 rArr ((x-2)(x+1))/((x(x+2)) lt 0 x in (-2,-1) cup (0,2)`