Question

Find the values of ‘a’ for which f(x) = `sqrt3 sin x - cos x - 2 ax + b` is decreasing on R.

Answer 1

Given: f(x) = `sqrt3 sin x - cos x - 2 ax + b`

Differentiate w.r.t. x,

`f'(x) = sqrt3 cos x + sin x - 2a`

For decreasing function

f'(x) ≤ 0

`sqrt3 cos x + sin x - 2a ≤ 0`

`sqrt3 cos x + sin x ≤ 2a`

`2[sqrt3/2  cos x + 1/2  sin x] ≤ 2a`

`2[1/2  sin x + sqrt3/2  cos x] ≤ 2a`

`2[sin  pi/6 sin x + cos  pi/6 cos x] ≤ 2a`

`2sin(x + pi/6) ≤ 2a`

`sin(x + pi/6) ≤ a`

Maximum value of `sin(x + pi/6)` is 1.

∴ 1 ≤ a

So, a ≥ 1