Proving that a function is bounded

Question

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.

Have you tried letting $z=y^2$? – Kimball Jun 08 '11 at 00:01 — Kimball, Jun 08 '11 at 00:01

score 7 · Accepted Answer · edited Dec 06 '11 at 10:35

7

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

edited Dec 06 '11 at 10:35

t.b.

78,116

answered Jun 08 '11 at 00:05

Ross Millikan

374,822

+1. This is a really elegant hint. You might want to replace $>$ with $\ge$ – Jun 08 '11 at 00:06
@Sivaram: I noticed it about the same time. Fixed. – Ross Millikan Jun 08 '11 at 00:08
1

@Leonardo Once you have solved the problem using the hint, you might want to look into the arithmetic-geometric inequality, which is a generalization of the idea behind the hint, and a useful tool to have in your arsenal. – Aaron Jun 08 '11 at 00:12
@Aaron I'm well aware of that, thanks.
Great hint, the answer comes practically for free.
– Leonardo Fontoura Jun 08 '11 at 00:29
It took me a second to see what you were getting at, but I really appreciate this hint. Nicely constructed. – davidlowryduda Jun 08 '11 at 01:18

Proving that a function is bounded

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