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Home » "From Tiny Drops to Epic Battles: Discover the Ultimate Water Pokémon Today! - Get link 4share

"From Tiny Drops to Epic Battles: Discover the Ultimate Water Pokémon Today! - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: Any three consecutive integers include at least one multiple of 2 and one multiple of 3. Thus, the product is divisible by $2 imes 3 = 6$. Additionally, among three consecutive numbers, there is a multiple of 2 and another even number, ensuring divisibility by $2^2 = 4$ if the sequence includes two even numbers. However, the guaranteed minimum is $3! = 6$. But more precisely, the product of three consecutive integers is divisible by $6$, but not necessarily higher (e.g., $1 imes 2 imes 3 = 6$). Thus, the answer is $oxed{6}$.Question: An ichthyologist models the seasonal variation in coral reef fish populations using the function $ f(x) = \sin^2 x + \cos^2(2x) $. Find the minimum value of $ f(x) $ over all real numbers $ x $.

📌 f(x) = \sin^2 x + \cos^2(2x)

📌 We use trigonometric identities to simplify and analyze the expression. Recall:

📌 \sin^2 x = rac{1 - \cos 2x}{2}, \quad \cos^2(2x) = rac{1 + \cos 4x}{2}

📌 f(x) = rac{1 - \cos 2x}{2} + rac{1 + \cos 4x}{2} = rac{1}{2} - rac{\cos 2x}{2} + rac{1}{2} + rac{\cos 4x}{2} = 1 - rac{\cos 2x}{2} + rac{\cos 4x}{2}

📌 f(x) = 1 + rac{1}{2}(\cos 4x - \cos 2x)

📌 Now, use the identity for $ \cos 4x - \cos 2x $:

📌 \cos A - \cos B = -2 \sin\left(rac{A+B}{2}

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