Suppose the function $f(x,y,z)=xyz$ is defined for $x+y+z=7$, $x,y,z \ge 0$. What is the range of $f$?
Since $x,y,z \ge 0$, we have by the AM-GM inequality that
\[\sqrt[3]{xyz} \le \frac{x+y+z}{3} = \frac{7}{3}.\]Thus $xyz \le \frac{7^3}{3^3}=\frac{343}{27},$ with equality at $x = y = z = \frac{7}{3}.$

Also, $xyz \ge 0,$ so the range is $\boxed{\left[0,\frac{343}{27}\right]}$.