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Home » The sum of the five terms is: - Blask

The sum of the five terms is: - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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。

📌 \cos x + \sec x = \cos x + \frac{1}{\cos x}, \quad \sin x + \csc x = \sin x + \frac{1}{\sin x}

📌 \left(\cos x + \frac{1}{\cos x}\right)^2 + \left(\sin x + \frac{1}{\sin x}\right)^2

📌 \left(\cos x + \frac{1}{\cos x}\right)^2 =

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\Delta h = 12 - \sqrt{120} = 12 - 2\sqrt{30}
But compute exactly:
\sqrt{120} = \sqrt{4 \cdot 30} = 2\sqrt{30}
So decrease in height is \( 12 - 2\sqrt{30} \) meters.
However, the problem asks for how far the top slides down — which is t...
- 2\sqrt{30}
But this is not simplified numerically.
Alternatively, is there a better way?
But 2√30 is exact.
However, check if 5-12-13 is correct — yes.

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