How to solve this problem using Jensen's inequality? This is the task to find the minimum  value of $x^5 + y^5 + z^5 - 5xyz$
where $x,y$ and $z$ are any positive numbers.
I know that I may use this inequality:
$$(t_1\cdot t_2\cdot t_3\cdots t_n)^{\frac{1}{n}} \leq \frac{t_1+t_2+t_3+\cdots +t_n}{n}$$

By the inequality we have that $\frac{x^5 + y^5 + z^5 + 1 + 1}{5} \ge \sqrt[5]{x^5y^5z^5\cdot1 \cdot 1}$. Hence $x^5 + y^5 + z^5 + 2 \ge 5xyz$
$$x^5 + y^5 + z^5 - 5xyz = x^5 + y^5 + z^5 + 2 - 5xyz - 2 \ge 5xyz - 5xyz - 2 = -2$$
It's obtained for $x=y=z$
