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Home » From Garage Drop to Home in California – Hamer Toyota’s Latest Triple Win! - Midis

From Garage Drop to Home in California – Hamer Toyota’s Latest Triple Win! - Midis

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 Solution: The minimum occurs at $ x = -\frac{b}{2a} = -\frac{-20}{2(5)} = 2 $. Verify $ W(2) = 5(4) - 20(2) + 45 = 20 - 40 + 45 = 25 $. The smallest $ x $ is $ \boxed{2} $.Question: A hydrologist models groundwater flow with the function $ f(x) = (\sec x + \cos x)^2 + (\csc x - \sin x)^2 $ for $ 0 < x < \frac{\pi}{2} $. Find the minimum value of $ f(x) $.

📌 Solution: We are given the function

📌 f(x) = (\sec x + \cos x)^2 + (\csc x - \sin x)^2

📌 for $ 0 < x < \frac{\pi}{2} $. We aim to find its minimum value.

📌 First, expand both squared terms:

📌 Add the two expressions:

📌 f(x) = \sec^2 x + \cos^2 x + 2 + \csc^2 x + \sin^2 x - 2

📌 f(x) = \sec^2 x + \csc^2 x + (\cos^2 x + \sin^2 x) + 2 - 2 = \sec^2 x + \csc^2 x + 1

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