Question 92
Prove that if $0 \lt x_1 \le x_2 \le \dots \le x_n$ and $k \in \mathbb{N}$ then the inequality
$$
\sqrt[k]{x_1^k + x_n^k - \frac{x_1^k + \dots + x_n^k}{n}} \geq x_1 + x_n - \frac{x_1 + \dots + x_n}{n} \text{ holds.}
$$
$$
\sqrt[k]{x_1^k + x_n^k - \frac{x_1^k + \dots + x_n^k}{n}} \geq x_1 + x_n - \frac{x_1 + \dots + x_n}{n} \text{ holds.}
$$