For any number $c \in(0,1)$, find the minimum and maximum values of

$$\sum_{i=1}^{n} x_{i}^{c}$$

subject to $\sum_{i=1}^{n} x_{i}=1, x_{1}, \ldots, x_{n} \geqslant 0$. Find all the points $\left(x_{1}, \ldots, x_{n}\right)$ at which the minimum and maximum are attained. Justify your answer.