|a^4 + b^4| = \left| \frac{1}{2} \right| = \boxed{\frac{1}{2}}
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- 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}
- Solution: To minimize
- under the constraint $ x + y + z = 1 $, we apply the **Cauchy-Schwarz Inequality** in the following form:
- Equality occurs when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we have $ x = y = z = \frac{1}{3} $.
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