Find global minima of nonlinear, scalar, positive function numerically?

Question

Let us consider a real, smooth vector function $g(x): \mathbb{R}^n\rightarrow \mathbb{R}$ which is globally increasing, e.g. $\exists r > 0$ for which $g(x)$ with $||x|| > r$ is monotonically increasing with respect to $||x||$. Then the roots of $g(x)$ are necessarily inside of the $r$-sphere; I wonder how to calculate them efficiently? Right now I find $r$, diskretize the sphere, find all $x$ below a threshold $||x|| < \epsilon$ and discretize these areas further, but this method is possibly unstable and probably there is a more intelligent solution?