Consider a $3D$ figure represented by $xyz^2=2$. Then what is its minimum distance from its origin?
What I try:
Given $xyz^2=2$ and I have to find minimum of $x^2+y^2+z^2$
Let $x^2+y^2+z^2=k^2$ and $xy(k^2-x^2-y^2)=2$
How do I solve it? Please help
2026-03-26 21:25:47.1774560347
Minimum distance of the curve $xyz^2=2$ from its origin
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By AM-GM $$x^2+y^2+z^2=x^2+y^2+2\left(\frac{z^2}{2}\right)\geq4\sqrt[4]{x^2y^2\left(\frac{z^2}{2}\right)^2}=4.$$ The equality occurs for $x=y=\frac{z}{\sqrt2}$ and $xyz^2=2,$ which says that the distance is $2$.