$y=\sqrt{x^2-12x+93}=\sqrt{x^2-12x+36+57}=$

$=\sqrt{\left(x-6\right)^2+57}$

$\begin{array}{l}\sqrt{(x-6)^2+57}\geq\sqrt{57}\\(x-6)^2+57=57\\(x-6)^2=0\\x=6\end{array}$