<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] You may convert it to become a function of two variables, the find the minimum value of that function. To do this, use
$$ 
z = \sqrt{1-(x^{2}+y^{2})} 
$$
And substitute this in 
$$ \frac{x}{\sqrt{1-x^{2}}} + \frac{y}{\sqrt{1-y^{2}}}  + 
\frac{z}{\sqrt{1-z^{2}}}$$
so the function will be
$$
f(x,y) =  \frac{x}{\sqrt{1-x^{2}}} + \frac{y}{\sqrt{1-y^{2}}}  + 
\frac{ \sqrt{1-(x^{2}+y^{2})} }{ \sqrt{x^{2}+y^{2}} } 
$$
Use partial derivatives to find the minimum value..?
</s>