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Home » \frac1a^2 + \frac1b^2 \geq \frac2ab - AMAZONAWS

\frac1a^2 + \frac1b^2 \geq \frac2ab - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 Therefore, the number of distinct sequences is $ \boxed{1260} $.Question: A technology consultant models data flow efficiency using the function $ f(\theta) = \sin^4 \theta - 4\sin^2 \theta \cos^2 \theta + \cos^4 \theta $. Find the minimum value of $ f(\theta) $ over all real $ \theta $.

📌 Solution: We begin by rewriting $ f(\theta) $ using trigonometric identities. Observe:

📌 f(\theta) = \sin^4 \theta - 4\sin^2 \theta \cos^2 \theta + \cos^4 \theta.

📌 Group the first and last terms:

📌 f(\theta) = (\sin^4 \theta + \cos^4 \theta) - 4\sin^2 \theta \cos^2 \theta.

📌 Recall that:

📌 \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta.

📌 Substitute into $ f(\theta) $:

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