If the value of the determinant […] is positive, then

We have,

$Δ=\begin{vmatrix}a&1&1\\1 &b& 1\\1&1&c\end{vmatrix}=abc - (a+b+c) +2$

$∴Δ>0$

$⇒abc +2> a+b+c$

$⇒abc+2>3(abc)^{1/3}$   $\left[∵A.M.>G.M.⇒\frac{a+b+c}{3}>(abc)^{1/3}\right]$

$x^3+2> 3x$, where $x =(abc)^{1/3}$

$⇒x^3-3x+2>0$

$⇒(x-1)^2 (x+2) > 0$

$⇒ x+2>0⇒ x>-2⇒ (abc)^{1/3} >-2 ⇒ abc >-8$