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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Expand both sides:

📌 - 12y + 4y^2 \leq y^2 - 4y + 4

📌 4y^2 - 12y + 9 \leq y^2 - 4y + 4

📌 3y^2 - 8y + 5 \leq 0

📌 Solve quadratic: discriminant = \(64 - 60 = 4\), roots:

📌 y = rac{8 \pm 2}{6} = rac{10}{6} = rac{5}{3},\quad rac{6}{6} = 1

📌 So \(1 \leq y \leq rac{5}{3}\)

📌 Check if all values are attainable: at \(\sin x = 1\), \(f = rac{2+3}{1+2} = rac{5}{3}\), at \(\sin x = -1\), \(f = rac{-2+3}{-1+2} = rac{1}{1} = 1\). Intermediate values follow continuously.

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