Let $g\left(x\right)=a{x}^2+bx+c,a,b,c\in N$ and satisfies ${\int }_0^{1}g\left(x\right)dx=\frac{11}{6}.$ Let f (x) be a continuous and derivable function in $(x_1,x_2).$ If $f\left(x\right).f\text{'}\left(x\right)\ge x\sqrt{1-{\left(f\left(x\right)\right)}^4}$ and $\underset{x\to x_1^{+}}{\lim }{\left(f\left(x\right)\right)}^2=1$ and $\underset{x\to x_2^{-}}{\lim }{\left(f\left(x\right)\right)}^2=\frac{1}{2},$ then the minimum value of $[x_1^2-x_2^2]$ is equal to, (where [.] denotes greatest integer function.)