Let f x = sin 4 ⁡ x + cos 4 ⁡ x . Then, f is an increasing function in the interval:

Step-by-step solution

f x = sin 4 ⁡ x + cos 4 ⁡ x f ′ x = 4 sin 3 ⁡ x cos ⁡ x + 4 cos 3 ⁡ x - sin ⁡ x = 4 sin ⁡ x cos ⁡ x sin 2 ⁡ x - cos 2 ⁡ x =   - 2 sin ⁡ 2 x cos ⁡ 2 x =   - sin ⁡ 4 x f x , is increasing when f ′ x > 0 ⇒ - sin ⁡ 4 x > 0 ⇒ sin ⁡ 4 x < 0 ⇒ π < 4 x < 2 π ⇒ π 4 < x < π 2 ⇒ x   ∈ π 4 , π 2