If m and M are the minimum and the maximum values of 4 + 1 2 sin 2 ⁡ 2 x - 2 cos 4 ⁡ x , x ∈ R , then M - m is equal to:

Step-by-step solution

Given expression is 4 + 1 2 sin 2 ⁡ 2 x - 2 cos 4 ⁡ x =  4 + 2 1 - cos 2 ⁡ x cos 2 ⁡ x - 2 cos 4 ⁡ x Creating perfect squares, we get = - 4   cos 2 ⁡ x - 1 4 2 - 17 16   Since, 0   ≤ cos 2 ⁡ x   ≤ 1 ⇒ - 1 4   ≤ cos 2 ⁡ x - 1 4   ≤ 3 4 ⇒ 0   ≤ cos 2 ⁡ x - 1 4 2   ≤ 9 16 ⇒ - 17 16   ≤ cos 2 ⁡ x - 1 4 2 - 17 16   ≤ 9 16 - 17 16 ⇒ 17 4