Let f(x,y)=((x^(2)y^(2))/(x^(2)+y^(2))), classify the behavior of f near the critical point (0,0).

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- Feb 7th 2007, 02:39 AMbobby77multivarible extremum point
Let f(x,y)=((x^(2)y^(2))/(x^(2)+y^(2))), classify the behavior of f near the critical point (0,0).

- Feb 7th 2007, 06:03 AMCaptainBlack
I don't know how you are supposed to approach this on your course,

but the following works:

Along the $\displaystyle x \,$ and $\displaystyle y\,$ axes the function is a constant equal to zero. Along

any other ray through the origin we may put $\displaystyle y=\lambda x\,$, when:

$\displaystyle f(x,\lambda x)=\frac{\lambda^2 x^2}{1+\lambda^2}\,$

which has a quadratic like mininum at $\displaystyle x=0\,$.

A surface plot shows what this looks like (see attachment)

(this assumes that we give the function a value of $\displaystyle 0\,$ at $\displaystyle x=y=0\,$)

RonL - Feb 7th 2007, 06:38 AMThePerfectHacker
- Feb 7th 2007, 07:11 AMCaptainBlack
- Feb 7th 2007, 07:17 AMThePerfectHacker
- Feb 7th 2007, 10:29 PMCaptainBlack