The difference between the maximum and minimum values of the function f x = sin 3 ⁡ x - 3 sin ⁡ x , ∀ x ∈ 0 , π 6 is

Step-by-step solution

Let, sin ⁡ x = t ∴   If   x ∈ 0 , π 6   ⇒ t ∈ 0 , 1 2 Now, f t = t 3 - 3 t ⇒ f ' t = 3 t 2 - 3 = 3 t - 1 t + 1 Thus, f t is decreasing ∀ t ∈ 0 , 1 2 ∴ f t m a x = f 0 = 0 f t m i n = f 1 2 = 1 8 - 3 2 = - 11 8 ∴ f m a x - f m i n = 11 8