Concavity and Inflection Points

Let $f(x)=\frac{1}{3}x^3-2x+1$. Consider the following statements.

I. A potential inflection point exists at $x=0$ because $f''(x)=2x$ and $f''(0)=0$.

II. The graph of $f$ is concave down for $x<0$ and concave up for $x>0$.

III. The function $f$ has a relative minimum at $x=0$.

Which of the above statements are true?