Least value of algebraic expression

Question

Least value of $x^{2017}+y^{2017}+z^{2017}-2017xyz$, where $x,y,z\geq 0$

Try: I am trying to solve it using the arithmetic-geometric inequality

$$\frac{x^{2017}+y^{2017}+z^{2017}}{3}\geq (xyz)^{\frac{2017}{3}}$$

Could some help me to solve it , thanks

Answer 1

Consider the AM-GM $$x^{2017}+y^{2017}+z^{2017}+\underbrace{1+1+\cdots+1}_{2014 \text{ terms}} \geqslant 2017xyz$$ $$\implies x^{2017}+y^{2017}+z^{2017}-2017xyz \geqslant -2014$$ and note equality is possible in the AM-GM when $x=y=z=1$

Answer 2

Hint: Using your result from AM-GM, $$x^{2017}+y^{2017}+z^{2017}-2017xyz\ge3(xyz)^{\frac{2017}3}-2017xyz\,.$$ Now set $w=xyz$ and minimize $3w^{\frac{2017}3}-2017w$ on the RHS.