<s>[INST] Minimum of $\frac{x}{\sqrt{1-x^2}}+\frac{y}{\sqrt{1-y^2}}+\frac{z}{\sqrt{1-z^2}}$ Let $0\leq x,y,z<1$ and $x^2+y^2+z^2=1$. What is the minimum value of
$$\frac{x}{\sqrt{1-x^2}}+\frac{y}{\sqrt{1-y^2}}+\frac{z}{\sqrt{1-z^2}}?$$
From the condition, the point $(x,y,z)$ lies on a sphere with radius $1$. At the equality point, all three variables are equal to $1/\sqrt{3}$, and the value taken is $3/\sqrt{2}$. If one variable is $0$ and the other two are equal, the two variables are equal to $1/\sqrt{2}$, and the value taken is $2$.
 [/INST] By AM-GM $$\sum_{cyc}\frac{x}{\sqrt{1-x^2}}=\sum_{cyc}\frac{2x^2}{2\sqrt{x^2(1-x^2)}}\geq\sum_{cyc}\frac{2x^2}{x^2+1-x^2}=2.$$
The equality occurs for $z=0$ and $x=y=\frac{1}{\sqrt2},$ which says that $2$ is a minimal value.
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