Let \(x_1,x_2,\ldots,x_n\) be nonnegative real numbers. Show that
\(\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i \).