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Home » The product of the roots is \(\boxed6\). - Deep Underground Poetry

The product of the roots is \(\boxed6\). - Deep Underground Poetry

📅 May 4, 2026 👤 scraface

May 04, 2026

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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^...
Now, since \(0 < x < \frac{\pi}{2}\), \(u = \sin x + \cos x = \sqrt{2}...
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 \(v \in (1, 2]\).
Take derivative:
g'(v) = \frac{4(v - 1)^2 - 4v \cdot 2(v - 1)}{(v - 1)^4} = \frac{4(v -...
For \(v > 1\), \(g'(v) < 0\), so \(g(v)\) is decreasing on \((1, 2]\)....

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