Think periodically

Algebra Level 5

Consider the functions \(f(x)\) such that \[f(x+2)+f(x)+f(x-2)=f(x+1)+f(x-1)\] for all real \(x\). Find the smallest positive number \(P\) that is a period for all such functions \(f(x).\)

For extra credit, give an example of a non-zero function \(f(x)\) of this form that has a fundamental period less than \(P.\)

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