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How would I show that the origin of the function $$g(x,y) = x^6 - y^6x^2$$ is a saddle point?

I worked out the Hessian matrix and at $(0,0)$ this matrix is null. Where do I go from here?

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1 Answer 1

up vote 3 down vote accepted

$g(x,y)=x^2(x^4-y^6)$ and $g(x,a|x|^{2/3})=x^2(x^4-ax^4)=x^2(1-a)x^4$ for every $a$. You can choose $a<1$ to see that $g$ take positive values in a neighborhood of $(0,0)$, and $a>1$ for negative values.

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