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Algebra Level 4

Let \({ x }_{ 1 }, { x }_{ 2 }, { x }_{ 3 }, { x }_{ 4 }, { x }_{ 5 }, { x }_{ 6 }\) be real numbers such that \({ x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 }+{ x }_{ 4 }+{ x }_{ 5 }+{ x }_{ 6 }=0\) and \({ x }_{ 1 }^{ 2 }+{ x }_{ 2 }^{ 2 }+{ x }_{ 3 }^{ 2 }+{ x }_{ 4 }^{ 2 }+{ x }_{ 5 }^{ 2 }+{ x }_{ 6 }^{ 2 }=6\). Find the maximum value of \({ x }_{ 1 }{ x }_{ 2 }{ x }_{ 3 }{ x }_{ 4 }{ x }_{ 5 }{ x }_{ 6 }\).

Write your answer to 2 decimal places.

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Source: A problem in Russia, 2005.

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