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Home » Use exponential growth: \( 120 \times (1.15)^4 \). - Parker Core Knowledge

Use exponential growth: \( 120 \times (1.15)^4 \). - Parker Core Knowledge

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Define $ g(s) = 1 + 2s + \frac{1}{s} $, for $ 0 < s \leq \frac{1}{2} $.

📌 g'(s) = 2 - \frac{1}{s^2}

📌 Set $ g'(s) = 0 \Rightarrow 2 = \frac{1}{s^2} \Rightarrow s^2 = \frac{1}{2} \Rightarrow s = \frac{1}{\sqrt{2}} \approx 0.707 > \frac{1}{2} $, not in domain.

📌 So minimum occurs at endpoint $ s = \frac{1}{2} $:

📌 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:

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