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If f(x) satisfies the relation f(x)+f(x+...

If `f(x)` satisfies the relation `f(x)+f(x+4)=f(x+2)+f(x+6)` for all`x ,` then prove that `f(x)` is periodic and find its period.

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Given `f(x)+f(x+4)=f(x+2)+f(x+6) " (1) " `
Replace x by `x+2`. Then
`f(x+2)+f(x+6)=f(x+4)+f(x+8) " (2)" `
From (1) and (2), we have `f(x)=f(x+8).`
Hence, `f(x)` is periodic with period 8.
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