KöMaL 2007 A422

Let $x_1$,$x_2$,...,$x_n$,$x_{n+1}$ be positive real numbers with $x_1$+$x_2$+...+$x_n$=$x_{n+1}$. Prove that $\sum_{i=1}^n\sqrt{x_i(x_{n+1}-x_i)} \le \sqrt{\sum_{i=1}^n x_{n+1}(x_{n+1}-x_i)}.$