Thus, the minimum value is \(\boxed{\frac{3}{2}}\).

February 22, 2026

$Thus, the minimum value is \(\boxed{\frac{3}{2}}\).$

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\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{3}{2}
Equality holds when \( x = y = z \). Since \( x + y + z = 6 \), set \( x = y = z = 2 \).
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}
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