If \(x^{3}-2 x^{2}-9 x+18=0\) and \(A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|\), then the maximum value of \(A\) is

Given, \(x^{3}-2 x^{2}-9 x+18=0\)
and \(A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|\)
\(x^{3}-2 x^{2}-9 x+18=0\)
\(\Rightarrow \quad x^{2}(x-2)-9(x-2)=0\)
\(\Rightarrow \quad\left(x^{2}-9\right)(x-2)=0\)
\(\Rightarrow \quad(x-3)(x+3)(x-2)=0\)
\(\therefore \quad x=2,3,-3\)
\(A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|\)
\(=1(9 x-48)-2(36-42)+3(32-7 x)\)
\(=9 x-48+12+96-21 x\)
\(=-12 x+60\)
\(A(\) when \(x=2)=-12 \times 2+60=36\)
\(A(\) when \(x=3)=-12 \times 3+60=24\)
\(A(\) when \(x=-3)=-12 \times(-3)+60=96\)
More value of \(A\) at \(x=-3\) is \(96 .\)