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Home » Solution: Let us define a function $ f(u) $ such that - Blask

Solution: Let us define a function $ f(u) $ such that - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Let \(u = \sin x + \cos x\). Then:

📌 u^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + 2\sin x \cos x

📌 \Rightarrow \sin x \cos x = \frac{u^2 - 1}{2}

📌 f(x) = \left(\frac{u}{\frac{u^2 - 1}{2}}\right)^2 = \left(\frac{2u}{u^2 - 1}\right)^2 = \frac{4u^2}{(u^2 - 1)^2}

📌 Now, since \(0 < x < \frac{\pi}{2}\), \(u = \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\), so \(u \in (1, \sqrt{2}]\) (maximum \(\sqrt{2}\) at \(x = \frac{\pi}{4}\)).

📌 Let \(v = u^2\), so \(v \in (1, 2]\). Then:

📌 f(x) = \frac{4v}{(v - 1)^2}

📌 Let \(g(v) = \frac{4v}{(v - 1)^2}\). Find maximum on \(v \in (1, 2]\).

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