Sufficiency and MLE — power distribution

Let $x_1,x_2,\dots,x_n$ be a random sample from a population with probability density function $f(x;\theta)=\theta x^{\theta-1}$; $0<x<1$, $\theta>0$. Which of the following statements are correct?

I. $\prod_{i=1}^n x_i$ is a sufficient estimator of $\theta$.

II. $\sum_{i=1}^n x_i$ is a sufficient estimator of $\theta$.

III. $-\dfrac{n}{\sum_{i=1}^n\log x_i}$ is a maximum likelihood estimator of $\theta$.