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Therefore, the horizontal distance is - Decision Point

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2

📌 We already know $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $, so

📌 a^2b^2 = (ab)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}

📌 Substituting into the identity:

📌 a^4 + b^4 = 1^2 - 2 \cdot \frac{1}{4} = 1 - \frac{1}{2} = \frac{1}{2}

📌 |a^4 + b^4| = \left| \frac{1}{2} \right| = \boxed{\frac{1}{2}}

📌 Solution: To minimize

📌 under the constraint $ x + y + z = 1 $, we apply the **Cauchy-Schwarz Inequality** in the following form:

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