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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 |
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Answer» 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. |
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