If f x = x 3 2 x 2 + 1 1 + 3 x 3 x 2 + 2 2 x x 3 + 6 x 3 − x 4 x 2 − 2 for all x ∈ ℝ , then 2 f 0 + f ' 0 is…

Given: $f\left(x\right)=\left|\begin{array}{ccc}{x}^3& 2x^2+1& 1+3x\\ 3x^2+2& 2x& x^3+6\\ x^3-x& 4& x^2-2\end{array}\right|$

$\Rightarrow f\left(0\right)=\left|\begin{array}{ccc}0& 1& 1\\ 2& 0& 6\\ 0& 4& -2\end{array}\right|$

$\Rightarrow f\left(0\right)=0+4+8$

$\Rightarrow f\left(0\right)=12$

$\Rightarrow f^{\text{'}}\left(x\right)=\left|\begin{array}{ccc}{x}^3& 2x^2+1& 1+3x\\ 3x^2+2& 2x& x^3+6\\ 3x^2-1& 0& 2x\end{array}\right|+\left|\begin{array}{ccc}{x}^3& 2x^2+1& 1+3x\\ 6x& 2& 3x^2\\ x^3-x& 4& x^2-2\end{array}\right|+\left|\begin{array}{ccc}3x^2& 4x& 3\\ 3x^2+2& 2x& x^3+6\\ x^3-x& 4& x^2-2\end{array}\right|$

$\Rightarrow f^{\text{'}}\left(0\right)=\left|\begin{array}{ccc}0& 1& 1\\ 2& 0& 6\\ -1& 0& 0\end{array}\right|+\left|\begin{array}{ccc}0& 1& 1\\ 0& 2& 0\\ 0& 4& -2\end{array}\right|+\left|\begin{array}{ccc}0& 0& 3\\ 2& 0& 6\\ 0& 4& -2\end{array}\right|$

$\Rightarrow f^{\text{'}}\left(0\right)=0-\left(0+6\right)+0+0+0+0+3\left(8\right)$

$\Rightarrow f^{\text{'}}\left(0\right)=-6+24=18$

$\Rightarrow 2f\left(0\right)+f^{\text{'}}\left(0\right)=42$