If f(x)=x32x2+11+3x3x2+22xx3+6x3−x4x2−2 for all x∈R, then 2f(0)- f' (0) is equal to

f(0)=|01120604−2|=12f'(x)=|3x24x33x2+22xx3+6x3−x4x2−2|+|x32x2+11+3x6x23x2x3−x4x2−2|+|x32x2+11+3x3x2+22xx3+63x2−102x|∴f'(0)=182f(0)−f'(0)=24−18=6

If f(x)=x32x2+11+3x3x2+22xx3+6x3−x4x2−2 for all x∈R, then 2f(0)- f' (0) is equal to

f(0)=|01120604−2|=12f'(x)=|3x24x33x2+22xx3+6x3−x4x2−2|+|x32x2+11+3x6x23x2x3−x4x2−2|+|x32x2+11+3x3x2+22xx3+63x2−102x|∴f'(0)=182f(0)−f'(0)=24−18=6

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If $f (x) = |\begin{matrix} x^{3} & 2 x^{2} + 1 & 1 + 3 x \\ 3 x^{2} + 2 & 2 x & x^{3} + 6 \\ x^{3} - x & 4 & x^{2} - 2 \end{matrix}|$ for all $x \in R$ , then $2 f (0) - f'$ (0) is equal to

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Detailed Solution

$\begin{array}{l} f (0) = | \begin{matrix} 0 & 1 & 1 \\ 2 & 0 & 6 \\ 0 & 4 & - 2 \end{matrix} | = 12 \\ f' (x) = | \begin{matrix} 3 x^{2} & 4 x & 3 \\ 3 x^{2} + 2 & 2 x & x^{3} + 6 \\ x^{3} - x & 4 & x^{2} - 2 \end{matrix} | + | \begin{matrix} x^{3} & 2 x^{2} + 1 & 1 + 3 x \\ 6 x & 2 & 3 x^{2} \\ x^{3} - x & 4 & x^{2} - 2 \end{matrix} | + | \begin{matrix} x^{3} & 2 x^{2} + 1 & 1 + 3 x \\ 3 x^{2} + 2 & 2 x & x^{3} + 6 \\ 3 x^{2} - 1 & 0 & 2 x \end{matrix} | \end{array}$