Fill in the blanks in each of the following

The values of a for which the function f (x) = sinx – ax + b increases on R are ______.

Given f (x) = sinx – ax + b

Now apply the derivative we get

Now applying the sum rule of differentiation and the differentiation of the constant term is 0 we get

Now applying the derivative, we get

f'(x)=cos x-a

Now given f(x) increases on R

⇒ f’(x)≥0 ∀ x

⇒ cos x-a≥0 ∀ x

⇒ cos x≥a ∀ x

This is possible when a≤-1

Hence a∈(-∞, -1]

The values of a for which the function f (x) = sinx – ax + b increases on R are (-∞, -1].