ext { If } f(x)=\left|\begin{array}{ccc} 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 \end{array} ight| ext { for all } x \in \mathbb{R} ext {, then } 2 f(0)+f^{\prime}(0) ext { is equal to }

Question

ext { If } f(x)=\left|\begin{array}{ccc} 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 \end{array}ight| 	ext { for all } x \in \mathbb{R} 	ext {, then } 2 f(0)+f^{\prime}(0) 	ext { is equal to }

Accepted Answer

Answer: C. 42 \begin{aligned} & f(0)=\left|\begin{array}{ccc} 0 & 1 & 1 \ 2 & 0 & 6 \ 0 & 4 & -2 \end{array}ight|=12 \ & f^{\prime}(x)=\left|\begin{array}{ccc} 3 x^2 & 4 x & 3 \ 3 x^2+2 & 2 x & x^3+6 \ x^3-x & 4 & x^2-2 \end{array}ight|+ \end{aligned} \begin{aligned} & \left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \ 6 x & 2 & 3 x^2 \ x^3-x & 4 & x^2-2 \end{array}ight|+ \ & \left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \ 3 x^2+2 & 2 x & x^3+6 \ 3 x^2-1 & 0 & 2 x \end{array}ight| \ & 	herefore f^{\prime}(0)=\left|\begin{array}{ccc} 0 & 0 & 3 \ 2 & 0 & 6 \ 0 & 4 & -2 \end{array}ight|+\left|\begin{array}{ccc} 0 & 1 & 1 \ 0 & 2 & 0 \ 0 & 4 & -2 \end{array}ight|+\left|\begin{array}{ccc} 0 & 1 & 1 \ 2 & 0 & 6 \ -1 & 0 & 0 \end{array}ight| \ & =24-6=18 \ & 	herefore 2 f(0)+f^{\prime}(0)=42 \end{aligned}

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$

Answer

Explanation

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