If f(x) = | cases x ^ 3 & 2 x ^ 2 + 1 & 1 + 3x 3 x ^ 2 + 2 & 2x & x...

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

⇒ f\left(0\right) = \left|\begin{cases}0 & 1 & 1 \\ 2 & 0 & 6 \\ 0 & 4 & - 2\end{cases}\right|

$⇒ f\left(0\right) = 0 + 4 + 8$

$⇒ f\left(0\right) = 12$

⇒ {f}^{'}\left(x\right) = \left|\begin{cases}{x}^{3} & 2{x}^{2} + 1 & 1 + 3x \\ 3{x}^{2} + 2 & 2x & {x}^{3} + 6 \\ 3{x}^{2} - 1 & 0 & 2x\end{cases}\right| + \left|\begin{cases}{x}^{3} & 2{x}^{2} + 1 & 1 + 3x \\ 6x & 2 & 3{x}^{2} \\ {x}^{3} - x & 4 & {x}^{2} - 2\end{cases}\right| + \left|\begin{cases}3{x}^{2} & 4x & 3 \\ 3{x}^{2} + 2 & 2x & {x}^{3} + 6 \\ {x}^{3} - x & 4 & {x}^{2} - 2\end{cases}\right|

⇒ {f}^{'}\left(0\right) = \left|\begin{cases}0 & 1 & 1 \\ 2 & 0 & 6 \\ - 1 & 0 & 0\end{cases}\right| + \left|\begin{cases}0 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 4 & - 2\end{cases}\right| + \left|\begin{cases}0 & 0 & 3 \\ 2 & 0 & 6 \\ 0 & 4 & - 2\end{cases}\right|

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

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

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