what is the critical point for f(x)=x^(1/3)?

A critical point occurs where the first derivative is either zero or does not exist.

First, find the first derivative: f'(x) = (1/3)·x^{-2/3}

The first derivative will be undefined when x = 0.

f(x) = x^(1/3)

f'(x) = (1/3)x^(-2/3) this derivative is undefined when x = 0......so this is a critical point

See the graph here :

https://www.desmos.com/calculator/w5bqt2akbh

As the graph shows......we have vertical tangent at x = 0