Why is not possible to find max and min with Lagrange multiplier in the corner of a set?

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I have a set $\{z<\sqrt{x^2+y^2}\}$ and I have to find the max and min distance between the set and a point. I don't understand why the Lagrange multiplier method doesn't work when the max and min are in the origin $(0,0,0)$. Are there another methods?