Question

If \( f(x)=\sin ^{2} x-3 \cos ^{2} x+2 a x-4 \) is increasing for all \( x \), then a is an element of

Answer 1

f(x) = sin²x − 3cos ²x + 2ax − 4

∴ f'(x) = 2 sin x cos x + 3 cos x sin x + 2a

= 5 sin x cos x + 2a

= 5/2 sin 2x + 2a

Given that f(x) is increasing for all values of x.

∴ f'(x) ≥ 0 for all values of x.

⇒ 5/2 sin 2x + 2a ≥ 0 for all values of x.

⇒ -5/2 + 2a ≥ 0 (∵ -1 ≤ sin 2x ≤ 1)

⇒ 2a ≥ 5/2

⇒ a ≥ 5/4

⇒ a ∈ [5/4, ∞) then given function f(x) is increasing for all values of x.