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📅 May 9, 2026 👤 scraface

May 09, 2026

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📌 \(x\) の係数の半分を 2 乗したものを加算および減算します:

📌 x^2 + 3x = \left(x + \frac{3}{2}\right)^2 - \frac{9}{4}

📌 f(x) = 4\left(\left(x + \frac{3}{2}\right)^2 - \frac{9}{4}\right) + 9 = 4\left(x + \frac{3}{2}\right)^2 - 9 + 9 = 4\left(x + \frac{3}{2}\right)^2

📌 項 \(4\left(x + \frac{3}{2}\right)^2\) は常に非負であり、\(\left(x + \frac{3}{2}\right)^2 = 0\) のときに最小値 0 を取ります。これは \(x = -\frac{3}{2}\) のときに起こります。

📌 \(x = -\frac{3}{2}\) を元の関数に代入します:

📌 f\left(-\frac{3}{2}\right) = 4\left(-\frac{3}{2} + \frac{3}{2}\right)^2 = 4 \cdot 0 = 0

📌 したがって、関数 \(f(x) = 4x^2 + 12x + 9\) の最小値は \(\boxed{0}\) です。

📌 問題: \(\left(\cos x + \sec x\right)^2 + \left(\sin x + \csc x\right)^2\) の最小値を求めよ。

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