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Home » $ f(x) = x $ for all $ x $ - Silent Sales Machine

$ f(x) = x $ for all $ x $ - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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Alternatively, polynomial division confirms:
Divide \( u^4 + 0u^3 + 0u^2 + 0u + 3 \) by \( u^2 - 2u + 2 \)
First: \( u^4 \div u^2 = u^2 \), multiply: \( u^2(u^2 - 2u + 2) = u^4...
Subtract: \( (u^4 + 0u^3 + 0u^2) - (u^4 - 2u^3 + 2u^2) = 2u^3 - 2u^2 \...
Bring down: \( +0u + 3 \), now divide \( 2u^3 \div u^2 = 2u \)
Multiply: \( 2u(u^2 - 2u + 2) = 2u^3 - 4u^2 + 4u \)
Subtract: \( (2u^3 - 2u^2 + 0u) - (2u^3 - 4u^2 + 4u) = 2u^2 - 4u \)
Now: \( 2u^2 \div u^2 = 2 \), multiply: \( 2(u^2 - 2u + 2) = 2u^2 - 4u...
Subtract: \( (2u^2 - 4u + 3) - (2u^2 - 4u + 4) = -1 \)
So remainder is \( -1 \)

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Set $ g'(s) = 0 \Rightarrow 2 = \frac{1}{s^2} \Rightarrow s^2 = \frac{...
So minimum occurs at endpoint $ s = \frac{1}{2} $:
g\left( \frac{1}{2} \right) = 1 + 2 \cdot \frac{1}{2} + \frac{1}{\frac...
Thus, the minimum value is $ \boxed{4} $.
Question: A philosopher of science considers three unit vectors $ \mat...
Solution: Since $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are unit vector...
\|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = \|\mathbf{a}\|^2 + \|\mat...
= 3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \m...
Each dot product $ \leq 1 $, and maximum occurs when $ \mathbf{a} = \m...
\|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = 3 + 2(1 + 1 + 1) = 3 + 6...

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