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Home » 现在令 $ x = -1.5 $： -

现在令 $ x = -1.5 $： -

📅 May 6, 2026 👤 scraface

May 06, 2026

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📌 g\left( \frac{1}{2} \right) = 1 + 2 \cdot \frac{1}{2} + \frac{1}{\frac{1}{2}} = 1 + 1 + 2 = 4

📌 Thus, the minimum value is $ \boxed{4} $.

📌 Question: A philosopher of science considers three unit vectors $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ in $ \mathbb{R}^3 $, representing distinct but balanced perspectives in a scientific model. Find the largest possible value of $ \|\mathbf{a} + \mathbf{b} + \mathbf{c}\| $.

📌 Solution: Since $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are unit vectors, we use the identity:

📌 \|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 + \|\mathbf{c}\|^2 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})

📌 = 3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})

📌 Each dot product $ \leq 1 $, and maximum occurs when $ \mathbf{a} = \mathbf{b} = \mathbf{c} $, then all dot products are 1:

📌 \|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = 3 + 2(1 + 1 + 1) = 3 + 6 = 9

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Unless — perhaps ratio of stone to bone tools means usage ratio?
But in context, likely the ratio is by count, and 3:1 is approximate,...
From usage, we computed s = 10, b = 8
But problem says ratio is 3:1 — so perhaps its a mistake.
Alternatively, perhaps ratio refers to something else.
But lets assume the usage-based equation is correct, and the 3:1 is mi...
From above, s = 10, b = 8 from usage data.
But the problem says ratio is 3:1 — so inconsistency.
Wait — perhaps the ratio of stone to bone tools is 3:1 by count, and w...
Unless total tools is not 18? But stated.

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