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B
≤0forallralθ
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C
≥0forallrealθ
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D
≤0onlywhenθ≤0
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Solution
The correct option is C≥0forallrealθ f(θ)=sinθ(sinθ+sin3θ)=(sinθ+3sinθ−4sin3θ)⋅sinθ=(4sinθ−4sin3θ)sinθ=sin2θ(4−4sin2θ)=4sin2θ(1−sin2θ)=4sin2θcos2θ=(2sinθcosθ)2=(sin2θ)2≥0
which is true for all θ. ∴f(θ)≥0∀θ∈R