Value of parameter a for which the function $f\left(x\right)=x^3+3(a-7)x^2+3\left(a^2-9\right)x-1$ has a positive point of maximum can be

$f \left(x\right) = x^3 + 3(a – 7) x^2 + 3(a^2 – 9)x – 1$

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

For $f\left(x\right)$ to have a positive point of maximum, both roots of $f^{\text{'}}\left(x\right)$ should be distinct and greater than $0$ 

$\therefore$Both roots of  ${\mathrm{x}}^2+2(\mathrm{a}-7)\mathrm{x}+\left({\mathrm{a}}^2-9\right)=0$ are greater than zero

$\left(i\right)$

$$\begin{gathered} \mathrm{D}>0\Rightarrow 4(\mathrm{a}-7{)}^2-4\left({\mathrm{a}}^2-9\right)>0 \\ \Rightarrow -14\mathrm{a}+58>0\Rightarrow \mathrm{a}<\frac{29}{7} \end{gathered}$$

$\left(ii\right)$${\mathrm{a}}^2-9>0\Rightarrow \mathrm{a}<-3or\mathrm{a}>3$

$\left(iii\right)$$\frac{-2(a-7)}{2·1}>0\Rightarrow a<7$

$\therefore a\in (-\infty ,-3)\cup \left(3,\frac{29}{4}\right).$