The least value of the product xyz for which the determinant $\left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|$ is non-negative is

$\begin{array}{rcl} & =& \mathrm{x}(\mathrm{yz}-1)-\mathrm{z}+1+1-\mathrm{y}\ge 0\\ & \Rightarrow & \mathrm{xyz}-(\mathrm{x}+\mathrm{y}+\mathrm{z})+2\ge 0\\ & \Rightarrow & \mathrm{xyz}-3{\left(\mathrm{xyz}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+2\ge 0 (\because \mathrm{AM}\ge \mathrm{GM})\\ \mathrm{Let} {\left(\mathrm{xyz}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} & =& \mathrm{a}\\ & \Rightarrow & {\mathrm{a}}^3-3\mathrm{a}+2\ge 0\\ & \Rightarrow & {(\mathrm{a}-1)}^2 (\mathrm{a}+2)\ge 0\\ & \Rightarrow & \mathrm{a}\ge -2\\ \therefore \mathrm{xyz}& \ge & -8\end{array}$

$\therefore$ Least value of xyz is -8