If the function $f\left(x\right)=x^3+3(a-7)x^2+3(a^2-9)x-1$ has a point of maximum at positive value of x then

$$\begin{gathered} f\left(x\right)=x^3+3(a-7)x^2+3(a^2-9)x-1 \\ f\text{'}\left(x\right)=3x^2+6(a-7)x+3(a^2-7) \end{gathered}$$

The roots of  positive and distinct which is possible if

$$\begin{gathered} \left(i\right) b^2-4ac>0\Rightarrow 6{(a-7)}^2-4\left(3\right)\left(3\right)(a^2-9)>0 \\ \Rightarrow a<\frac{29}{7} \\ \left(ii\right){\text{ Product of roots > 0 a}}^2-9>0 \\ \begin{array}{l}\left(iii\right)\text{\hspace{0.17em}Sum\hspace{0.17em}of\hspace{0.17em}roots\hspace{0.17em}}>0a-7<0\\ a<7\end{array} \end{gathered}$$

From i, ii, iii  $a\in (-\infty ,-3)\cup (3,\frac{29}{7})$