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dragon ball rage script - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Since $ \sqrt{13} \approx 3.605 $, $ 4 - \sqrt{13} \approx 0.395 > 0 $, so the expression never reaches zero.

📌 Therefore, the absolute value $ |2\sin x + 3\cos x + 4| $ achieves its minimum at the minimum of the continuous function on an interval where it is always positive.

📌 The minimum occurs when $ 2\sin x + 3\cos x $ is minimized, but since it's always greater than $ -\sqrt{13} $, and greater than $ -4 $, the expression inside is always positive.

📌 \min |2\sin x + 3\cos x + 4| = \min (2\sin x + 3\cos x + 4) = 4 - \sqrt{13} + 0? \quad \text{Wait — no: the minimum of the inner expression is $ 4 - \sqrt{13} \approx 0.395 $, so the minimum of the absolute value is just its minimum value because it's positive.

📌 \min |2\sin x + 3\cos x + 4| = 4 - \sqrt{13} \quad \text{if } 4 - \sqrt{13} > 0

📌 But $ 4 - \sqrt{13} > 0 $ since $ \sqrt{13} < 4 $, so yes.

📌 But wait: is this the minimum? Since the expression inside is always greater than $ 0 $, the absolute value is minimized exactly when $ 2\sin x + 3\cos x $ is minimized.

📌 \min (2\sin x + 3\cos x + 4) = 4 + \min(2\sin x + 3\cos x) = 4 + (-\sqrt{13}) = 4 - \sqrt{13}

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