Be negative!

Algebra Level 4

Consider the function\[\large f(x_1,\ldots ,x_n)=\sum_{k=1}^{n}x_k^2-x_1x_3-\sum_{k=2}^{n-1}x_kx_{k+1}\]

where \(n\geq 3\) and \(x_1,\ldots ,x_n\) are real numbers. Find the smallest value of \(n\) such that \(f\) attains some negative values.

If you come to the conclusion that no such \(n\) exists, enter 666.

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