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Find a function $f :\mathbb{R} \to \mathbb{R}$ satisfying that : $f(1)=1$ $f(x+y)=f(x)+f(y)+2xy$ $f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4} \hspace{5pt}\forall x \neq 0$

2 Answers 2

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Hint: $(x+y)^2=x^2+y^2+2xy{}$

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I do to here : let $g(x)=f(x)-x^2$ , at that time : $g(x+y)=g(x)+g(y)$ $g(1)=g(0)=0$ $g(\frac{1}{x})=\frac{g(x)}{x^{4}}$

I proved $g(x)\equiv 0$ with $ x \in \mathbb{Q}$ but proving $g(x) \equiv 0$ with $x \in \mathbb{R}$ is problem for me, can you help me more

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    @LevanDokite So you proved that $f(x)=x^2 , \ x \in \mathbb{R}$ has the wanted properties. If you want to know if this $f$ is unique with the above properties maybe you should ask another question.2012-11-11