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f(x)=tan^(-1)(sinx+cosx), x gt0 is alway...

`f(x)=tan^(-1)(sinx+cosx), x gt0` is always and increasing function on the interval

A

`(0,pi)`

B

`(0,(pi)/2)`

C

`(0,(pi)/4)`

D

`(0,(3pi)/4)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given `f(x)=tan^(-1)(sinx+cosx)`
`f'(x)=1/(1+(sinx+cosx)^(2))(cosx-sinx)`
`=1/(1+1+2sinxcosx)(cosx-sinx)`
For function to be increasing `f'(x)gt0`
`impliescosx-sinxgt0impliestanxlt1`
`:.` Required interval `=(0,(pi)/4)`
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