Let $f\left( x \right) =3{ x }^{ 2 }-7x+c$ where $c$ is variable cofficient with $x > \frac 7 6$.

Determine the value of $c$ such that $f\left( x \right)$ touches $g\left( x \right) =\frac { \sqrt { 12x+49-12c } +7 }{ 6 }$

Give your answer as $\lfloor c \rfloor$.

Given to me as challenge by me friend Lakshay kumar