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\( F_11 = 89 \) - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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Since $ \sqrt{13} \approx 3.605 $, $ 4 - \sqrt{13} \approx 0.395 > 0 $...
Therefore, the absolute value $ |2\sin x + 3\cos x + 4| $ achieves its...
The minimum occurs when $ 2\sin x + 3\cos x $ is minimized, but since...
\min |2\sin x + 3\cos x + 4| = \min (2\sin x + 3\cos x + 4) = 4 - \sqr...
\min |2\sin x + 3\cos x + 4| = 4 - \sqrt{13} \quad \text{if } 4 - \sqr...
But $ 4 - \sqrt{13} > 0 $ since $ \sqrt{13} < 4 $, so yes.
But wait: is this the minimum? Since the expression inside is always g...
\min (2\sin x + 3\cos x + 4) = 4 + \min(2\sin x + 3\cos x) = 4 + (-\sq...
And since this value is positive, the absolute value does not introduc...
Thus, the minimum value is $ \boxed{4 - \sqrt{13}} $.

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