Let for i=1,2,3, p_i(x) be a polynomial of degree 2 in x, p_i^′(x) and p_i^′ ′(x) be the first and second order derivatives of p_i(x) respectively.…

Let for $\displaystyle i=1,2,3, p_i(x)$ be a polynomial of degree $\displaystyle 2$ in $\displaystyle x, p_i^{\prime}(x)$ and $\displaystyle p_i^{\prime \prime}(x)$ be the first and second order derivatives of $\displaystyle p_i(x)$ respectively. Let, $$\mathrm{A}(x)=\left[\begin{array}{ccc} p_1(x) & p_1^{\prime}(x) & p_1^{\prime \prime}(x) \\ p_2(x) & p_2^{\prime}(x) & p_2^{\prime \prime}(x) \\ p_3(x) & p_3^{\prime}(x) & p_3^{\prime \prime}(x) \end{array}\right] $$ and $\displaystyle \mathrm{B}(x)=[\mathrm{A}(x)]^{\mathrm{T}} \mathrm{A}(x)$. Then determinant of $\displaystyle \mathrm{B}(x)$ :