$\text{ Let }{x}_0\text{ be the point of local maxima of }f\left(x\right)=\overrightarrow{a}·(\overrightarrow{b}\times \overrightarrow{c}),\text{ where }$

$\overrightarrow{\mathrm{a}}=x\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}},\overrightarrow{\mathrm{b}}=-2\hat{\mathrm{i}}+\mathrm{x}\hat{\mathrm{j}}-\hat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{c}}=7\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\mathrm{x}\hat{\mathrm{k}}\text{ . Then the value of }\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}·\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}·\overrightarrow{\mathrm{a}}$$\mathrm{at} \mathrm{x}={\mathrm{x}}_0\text{ is }$

$f\left(x\right)=\left|\begin{array}{ccc}x& -2& 3\\ -2& x& -1\\ 7& -2& x\end{array}\right|$

$\begin{array}{l}=x\left(x^2-2\right)+2(-2x+7)+3(4-7x)\\ =x^3-2x-4x+14+12-21x=x^3-27x+26\\ \Rightarrow f^{\text{'}}\left(x\right)=3x^2-27\Rightarrow f^{\text{'}}\left(x\right)=0\Rightarrow x=\pm 3\end{array}$

$$\begin{gathered} f"\left(x\right)=6x \\ \Rightarrow f"\left(-3\right)=-18<0 \\ \Rightarrow f\left(x\right) has local maximum at x=-3\Rightarrow x_0=-3 \end{gathered}$$

$\begin{array}{l}\overline{\mathrm{a}}=-3\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}};\overline{\mathrm{b}}=-2\hat{\mathrm{i}}-3\hat{\mathrm{j}}-\hat{\mathrm{k}};\overline{\mathrm{c}}=7\hat{\mathrm{i}}-2\hat{\mathrm{j}}-3\hat{\mathrm{k}}\\ \overline{\mathrm{a}}·\overline{\mathrm{b}}=6+6-3=9\\ \overline{\mathrm{b}}·\overline{\mathrm{c}}=-14+6+3=-5\mid \end{array}$

$\overline{c}.\overline{a}=-21+4-9=-26$

$\overline{a}.\overline{b}+\overline{b}.\overline{c}+\overline{c}.\overline{a}=-22$