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The interval in which the function f(...

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

A

`(-2,-1] cup (0,3]`

B

`(-2,-1 ] cup [0,3]`

C

`(-2,-1]cup [0,2]

D

`(-2,-1]cup ( 0,2]`

Text Solution

Verified by Experts

The correct Answer is:
D
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)`
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