The set of all values of a for which the function `f(x)=(a^2-3a+2)(cos^2\ x/4-sin^2\ x/4)+(a-1)x+sin1` does not possess critical points is (A) `[1,oo)` (B) `(0,1) uu (1,4)` (C) `(-2,4)` (D) `(1,3) uu (3,5)`

Options

A. `[1,oo]`
B. `(0,1)cup (1,4)`
C. (-2,4)
D. `(1,3)cup (3,5)`

Answer

B

Explanation

We have
`f(x)=(a^2-3a+2){cos^2""x/4-sin^2""x/4}+(a-1)x+sin1`
`rArr f(x)=(a-1)(a-2)cosx/2+(a-1)x+sin1`
`rArr f'(x)=-1/2(a-1)(a-2)sin""x/2+(a-1)`
`rArrf'(x)=(a-1){1-((a-2))/2sin""x/2}`
If f(x) does not process critical points , then
`f(x)ne 0 " for any "x inR`
`rArr (a-1){1-((a-2))/2sin""x/2}ne 0" for any "x in R`
`rArr ane1 and 1-((a-2))/2sin""x/2=0` must not have any solution in R
`rArr ane1and sin"x/2=2/(a-2)` is not solvable in R
`rArr ane and |2/(a-2)|gt1`
`rArr ane1 and |a-2| lt2`
`rArr ane1 and -2lta-2lt2`
`rArrane and 0ltalt4`
`rArr a in (0,1)cup(1,4)`