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$\begingroup$

This is one part of an exercise in my homework, which for some reason I can't think of any way to prove.

$\displaystyle f(x,y)=\frac{xy^2}{x^2+y^4}$, if $(x,y)\neq (0,0)$ and $0$ otherwise.

I'm trying to prove that this function is bounded. I have figured that I only need to prove it for $x\geq 0$, since $f(x,y)=-f(-x,y)$, but I can't really get around to why this is bounded near $0$.

Thanks for the help.

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    Have you tried letting $z=y^2$?2011-06-08

1 Answers 1

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Hint: $(x-y^2)^2 \ge 0$ ${}{}{}{}{}{}{}{}{}$

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    +1. This is a really elegant hint. You might want to replace $>$ with $\ge$2011-06-08
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    @Sivaram: I noticed it about the same time. Fixed.2011-06-08
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    @Leonardo Once you have solved the problem using the hint, you might want to look into the [arithmetic-geometric inequality](http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means), which is a generalization of the idea behind the hint, and a useful tool to have in your arsenal.2011-06-08
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    @Aaron I'm well aware of that, thanks. Great hint, the answer comes practically for free.2011-06-08
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    It took me a second to see what you were getting at, but I really appreciate this hint. Nicely constructed.2011-06-08