For x ∈ - π 2 , π 2 , the range of values of f x = 2 + sin ⁡ x + sin 3 ⁡ x + sin 5 ⁡ x … . . ∞ is

Step-by-step solution

We have, f x = 2 + sin ⁡ x cos 2 ⁡ x ⇒ f ′ x = cos 2 x cos x + sin x 2 cos x sin x cos 4 x = cos ⁡ x cos 2 ⁡ x + 2 sin 2 ⁡ x cos 4 ⁡ x = 1 + sin 2 ⁡ x cos 3 ⁡ x ⇒ f ′ x > 0 ∴ f x is an increasing function lim x → - π 2 ⁡ 2 + sin ⁡ x cos 2 ⁡ x = - ∞ and lim x → π 2 ⁡ 2 + sin ⁡ x cos 2 ⁡ x = ∞ ∴ Range = - ∞ , ∞