Find all functions \(f : \mathbb{R} \rightarrow \mathbb{R}\) such that \(f (x + y) + 2f (x - y) + f (x) + 2f (y) = 4x + y\) for all real \(x\) and \(y\).

For all x,
f(x)=0
I HAVE A PROOF BUT AM HAVING PROBLEM TO POST IT.
Hence, I GIVE THE STRATERGY
PLUG y=0,
You get f(x)=x-r where 2r=f(0)
Then plugging in for functions we get 8r=0
GIVING THE RESULT r=0 AND HENCE, f(x)=x for all x

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TopNewestSORRY, I MISTYPED IN THE FIRST LINE,f(x)=0 but it is f(x)=x

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For all x, f(x)=0 I HAVE A PROOF BUT AM HAVING PROBLEM TO POST IT. Hence, I GIVE THE STRATERGY PLUG y=0, You get f(x)=x-r where 2r=f(0) Then plugging in for functions we get 8r=0 GIVING THE RESULT r=0 AND HENCE, f(x)=x for all x

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