Distribution of the largest interval when partitioning [0,1] into n intervals with n-1 uniformly distributed points.

Question

Let $\lbrace x_1,x_2,...,x_{n-1}\rbrace$ be n-1 points that are independently uniformly distributed on interval [0,1]. They create n segments. What is the expected length of the largest of such n segments? What is the distribution?

A good if not complete solution here. Average length of the longest segment