Electric field due to a ring having charge Q and radius R at a distance x along its axis is E = $\frac{Qx}{4\pi\epsilon_0 (R^2+x^2)^\frac{3}{2}}$

E is maximum at a point where $\frac{dE}{dx}$ = 0 

$\Rightarrow x^2+R^2-3x^2$= 0

$\Rightarrow R^2 - 2x^2$ = 0 

x = $\frac{R}{\sqrt2}$