<s>[INST] What is the difference between the largest and smallest possible values of any of these numbers? 
If $x,y,z$ are real numbers such that $x+y+z = 5$ and $xy+yz+zx = 3$, what is the difference between the largest and smallest possible values of any of these numbers?

What is wrong with this approach?
We have that $(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+xz) = 25 \implies x^2+y^2+z^2 = 19$. Thus the points $(x,y,z)$ lie on a sphere with radius $\sqrt{19}$. Therefore the maximum value of any of these is $\sqrt{19}$ and the smallest is $-\sqrt{19}$ and the difference is $2\sqrt{19}$.
 [/INST] $(x,y,z)$ is not $\bf{three}$ points. It is a single point in three dimensional space, where $x,y,z$ are the coordinates.
Now think of a sphere of radius $\sqrt{19}$. If the $x$ coordinate is $\sqrt{19}$, then is it possible for any other coordinate to be $-\sqrt{19}$?
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