Is it true that the only solutions are $f(x)=0$ and $f(x)=x$?

Question

Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that

$$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did not come to any good result, but I think the solution should be difficult.

No, it is not true. There are two other constant solutions. – Andrés E. Caicedo Aug 26 '17 at 16:24 — Andrés E. Caicedo, Aug 26 '17 at 16:24
Let $f(x)=c$ be such that $c^2+c=1$. – Sándor Kovács Aug 26 '17 at 16:25 — Sándor Kovács, Aug 26 '17 at 16:25

score 2 · Accepted Answer · answered Aug 26 '17 at 16:26

2

No, the constant functions $$f=\frac{-1\pm\sqrt 5}{2}$$ are both solutions.

answered Aug 26 '17 at 16:26

Adrien Hardy

2,085

1

How do we show that these are the only solutions? – Noram Sadir Aug 26 '17 at 16:34
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@NoramSadir: That would be the next question! – Stefan Kohl Aug 26 '17 at 16:36

Is it true that the only solutions are $f(x)=0$ and $f(x)=x$?

1 Answers1