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If f(x)= (ab+b^2+1)x+oversetx underset0i...

If `f(x)= (ab+b^2+1)x+oversetx underset0int(cos^4theta+sin^4theta)" d"theta` is an incrasing function of x for all ` x in R` and `b in R` , b being independent of x then

A

`a in (0,sqrt6)`

B

`a in (sqrt6,sqrt6)`

C

`a in (-sqrt6,0)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
We have `f(x)= (ab+b^2+1)x+oversetx underset0int(cos^4theta+sin^4theta)" d"theta`
`rArr f'(X)= (ab+b^2+1)+cos^4x+sin^4x " for all " x in R` For f(x) to be increasing , we must have
`f'(x) gt0 " for all " x in R`
`rArr (ab+b^2+1)+(sin^4x+cos^4x)gt0 " for all " x in R`
`rArr ab+b^2+1+1/2gt0 `
`rArr 2ab+2b^2+3gt 0 " for all " b in R`
`rArr 2b^2+2ab+3gt0 " for all " b in R`
`4a^2-24lt0 rArra^2-6lt0 rArr -sqrt6ltaltsqrt6`
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