MathNet 2017-part-02 — 176 problems (EN–VI) Question 83

Real numbers $x$, $y$ and $z$ satisfy $x + y + z = 4$ and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{3}$. Find the largest and the smallest possible value of the expression $x^3 + y^3 + z^3 + xyz$.