Optimization | Part IB, 2004

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

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

subject to i=1nxi=1,x1,,xn0\sum_{i=1}^{n} x_{i}=1, x_{1}, \ldots, x_{n} \geqslant 0. Find all the points (x1,,xn)\left(x_{1}, \ldots, x_{n}\right) at which the minimum and maximum are attained. Justify your answer.

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