Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 20G (Optimization)

Each coordinate lies in $[0,1]$. Since $0<c<1$, $x_i^c\ge x_i$, with equality exactly when $x_i=0$ or $1$. Summing gives $f(\mathbf{x})\ge 1$. Equality under the simplex constraint forces exactly one coordinate to be one. Thus the minimum is $1$, attained exactly at the vertices of the probability simplex.

For the maximum, $t\mapsto t^c$ is strictly concave on $[0,\infty )$. The Jensen inequality givesStrict concavity makes equality possible exactly when all coordinates agree. Hence the extrema of a concave power sum on a simplex areFor $n=1$ the simplex is a single point, and both descriptions coincide.