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Question
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
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Solution
y = `(4sinθ)/(2 + cosθ) - θ`
∴ `dy/"dθ" = d/"dθ"[(4sinθ)/(2 + cosθ) - θ]`
= `d/"dθ"((4sinθ)/(2 + cosθ)) - d/"dθ"(θ)`
= `((2 + cosθ).d/"dθ"(4sinθ) - 4sinθ.d/"dθ"(2 + cos θ))/((2 + cosθ)^2) - 1`
= `((2 + cosθ)"(4cosθ) - (4sinθ)(0 - sinθ))/((2 + cosθ)^2) - 1`
= `(8cosθ + 4cos^2θ + 4sin^2θ)/(2 + cosθ)^2 - 1`
= `(8cosθ + 4(cos^2θ + sin^2θ))/(2 + cosθ)^2 - 1`
= `(8cosθ + 4)/(2 + cosθ)^2 - 1`
= `((8cos θ + 4) - (2 + cosθ)^2)/(2 + cosθ)^2`
= `(8cosθ + 4 - 4 - 4cosθ - cos^2θ)/(2 + cosθ)^2`
= `(4cosθ - cos^2θ)/(2 + cosθ)^2`
= `(cosθ(4 - cosθ))/(2 + cosθ)^2`
Since, `θ ∈ [0, pi/2], cos θ ≥ 0` Also, cos θ < 4
∴ 4 - cos θ > 0
∴ cos θ (4 - cosθ) ≥ 0
∴ `(cosθ(4 - cosθ))/(2 + cosθ^2)≥ 0`
∴ `dy/"dθ" ≥ 0 "for all" θ ∈[0, pi/2]`
Hence, y is an increasing function if `θ ∈[0, pi/2]`.
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