The value of ‘c’ so that for all real$\text{'}x\text{'}$ the vectors $\left(cx\overline{i}-6\overline{j}+3\overline{k}\right), x\overline{i}+2\overline{j}+2cx\overline{k}$ make an obtuse angle for

$$\begin{gathered} \cos \theta =\frac{\overline{a}\cdot \overline{b}}{\left|\overline{a}\right|\left|\overline{b}\right|}<0 \\ \Rightarrow \frac{\left(c^2-12+6cx\right)}{\sqrt{c^2{x}^2+36+9}\sqrt{x^2+4+4c^2{x}^2}}<0 \\ \Rightarrow {\mathrm{cx}}^2+6\mathrm{cx}-12<0 \\ \Rightarrow \mathrm{c}\left({\mathrm{x}}^2+6\mathrm{x}\right)<12 \\ \Rightarrow c<\frac{12}{x^2+6x}=\frac{12}{x^2+6x+9-9}=\frac{12}{(x+3{)}^2-9} \\ \Rightarrow \mathrm{c}<min\left[\frac{12}{(\mathrm{x}+3{)}^2-9}\right] \\ \therefore c<0 \end{gathered}$$