Question

The least value of 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:

Answer 1

$\left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|$ is non-negative

$\Rightarrow \left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|\ge 0$

$\Rightarrow x(yz-1)-1(z-1)+1(1-y)\ge 0$$\Rightarrow xyz-x-z+1+1-y\ge 0$

$\Rightarrow xyz+2\ge x+y+z$

So that minimum value of $x,y,z$ are $x=-2,y=-2,z=-2$ which satisfy the inequality

$\therefore xyz=-2\times -2\times -2=-8$