In the implicit function theorem, it is stated that (Analysis on Manifolds, Munkres, p74)
My question is that how can we find the largest $B$ s.t the function $g$ satisfies the conditions given in the theorem ?
I mean the theorem just states the existence of such $B$ as a neighbourhood of $x$; however, practically, how can we find the largest $B$ ?
Edit:
For example, since we do know the existance of such $B$, can we just compute $g$ and argue that the largest possible $C$ where $g$ is class of $C^r$ and $f(x,(g(x))) = 0$ is the largest possible $B$ ?
The size of this $B$ depends on many things: higher derivatives of $f$, the slope of the tangent plane to the graph of $g$ at $(a,b)$, what have you. Therefore it is almost impossible to give a cute answer. Consider as an example the equation $$f(x,y):=x^2+y^2-1=0\ .$$ At $(a,b):=(0,1)$ we have $g(x)=\sqrt{1-x^2} \quad (-1<x<1)$, but at $(a,b)=(0.9, 0.436)$ the function $g$ is only defined in an open interval of length $0.2$.
If you can "explicitly solve" $f(x,y)=0$ in the neighborhood of $(a,b)$ and are in possession of a finite expression for $g$ you don't need the implicit function theorem at all. Instead you can determine the radius of $B$ by inspecting this expression.