Find the set of all values of the parameter 'a' for which the function, `f(x) = sin 2x - 8(a + b)sin x + (4a^2 + 8a - 14)x` increases for all `x in R` and has no critical points for all `a in R.`

Options

A. `(-oo, -sqrt5,-2)`
B. `(1,oo)`
C. `(sqrt5,oo)`
D. None of these

Answer

B

Explanation

`f(x)=sin2x-8(a+1)sinx+(4a^(2)+8a-14)x`
`therefore" "f'(x)=2cos2x-8(a+1)cosx+4a^(2)+8a-14`
`=4cos^(2)x-2-8a cosx+4a^(2)+8a-14`
Now f(x) increase for all `x in R`
`rArr" "(cosx-(a+1))^(2)gt5`
`rArr" "cosx-(a+1)gt sqrt5 or cos x-(a+1)lt-sqrt5`
`rArr" "alt -sqrt5-1+cos x" "agt sqrt5-1+cosx`
`rArr" "alt-sqrt5-2" or" a gt sqrt5`