jmc

The value $x=f^{-1}(58)$ is the solution to $f(x)=58$. This means $$2x^3+4=58.$$Subtracting 4 gives $$2x^3=54.$$If we divide by 2 we get $$x^3=27,$$and the only value that solves this equation is $$x=\boxed{3}.$$