Putnam 1975 A3

Let $0<\alpha<\beta <\gamma\in \mathbb{R}$ . Let $K=\{(x,y,z)\in \mathbb{R}^{3}\;| x,y,z\geq 0 \text{and} x^{\beta}+y^{\beta}+z^{\beta}=1\}$ . Define $f:K\rightarrow \mathbb{R}, (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}$ . At what points of $K$ does $f$ assume its minimal and maximal values?