interesting fun eq

For discussing Olympiad Level Algebra (and Inequality) problems
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interesting fun eq

find all functions f such that for all real x,y $f:\mathbb{R}\rightarrow \mathbb{R},f(x^3+y^3)=x^2f(x)+yf(y^2)$

*Mahi*
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Re: interesting fun eq

The hardest part here is to prove $f$ is continuous.

Use $L^AT_EX$, It makes our work a lot easier!

Sakal
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Joined: Sun Jun 26, 2011 4:12 pm

Re: interesting fun eq

f(x)=cx is (c arbitary constant) [also,includes, c=0 means f(x)=0 ]
one solution I got..it's continuous .yet ,no other solution I got . trying.....

Corei13
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Re: interesting fun eq

Denote the statement by $P(x,y)$.
$P(x,0),P(0,x)$ imply that $f$ is a Cauchy equation and $f(x^2)=xf(x)$
Now, $(x+1)(f(x)+f(1))=(x+1)f(x+1)=f((x+1)^2)=f(x^2 + 2x + 1)$
$=xf(x)+2f(x)+f(1)$
So, $f(x)=xf(1)$.
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