Given $x+y+z=1$ Find Maximum value of $x^5y+y^5z+z^5x$

Question

$x,y,z$ are Non negative reals such that $x+y+z=1$ Find Maximum value of $$E=x^5y+y^5z+z^5x$$

The only idea i have is $E$ can be written as $f(x,y)$ and using partial differentiation for maxima...but too lengthy

Answer 1

Let $x\geq y\geq z$. Hence, by AM-GM $$x^5y+y^5z+z^5x\leq(x+z)^5y=5^5\left(\frac{x+z}{5}\right)^5y\leq5^5\left(\frac{5\cdot\frac{x+z}{5}+y}{6}\right)^6=\frac{5^5}{6^6}.$$ In the case $x\geq z\geq y$ we can use the same idea. $$x^5y+y^5z+z^5x\leq x^5z+y^5x+z^5y\leq(x+y)^5z\leq5^5\left(\frac{5\cdot\frac{x+y}{5}+z}{6}\right)^6=\frac{5^5}{6^6}$$ The equality occurs for $z=0$ and $y=\frac{x}{5}$, which gives the answer: $\frac{5^5}{6^6}$.