Given: f (x) = √3 sin X – cos X – 2ax + b
To show: the given function is decreasing in R.
Explanation: Given f (x) = √3 sin X – cos X – 2ax + b
Applying first derivative with respect to x, we get
But the derivative of sin X = cos x and that of cos x = -sin x, so
f' (x) = √3cos x-(-sin x)-2a
f' (x) = √3cos X+sin X-2a
Multiplying and dividing RHS by 2, we get
And we know, if f’(x)≤0, then f(x) is decreasing function.
Hence the given function is decreasing function in R.
