Evaluation Of Maximum Value Without Calculus $x$,$y$,$z$ are non-negative numbers.
$x+y+z=3$    
Find the maximum value of $~$ $x^{2}y+y^{2}z+z^{2}x$ $~$ without calculus.

Let $\{x,y,z\}=\{a,b,c\}$, where $a\geq b\geq c$.
Hence, by Rearrangement and AM-GM we obtain:
$$x^2y+y^2z+z^2x=x\cdot xy+y\cdot yz+z\cdot zx\leq a\cdot ab+b\cdot ac+c\cdot bc=$$
$$=b(a^2+ac+c^2)\leq b(a^2+2ac+c^2)=b(a+c)^2=4b\left(\frac{a+c}{2}\right)^2\leq$$
$$\leq4\left(\frac{b+\frac{a+c}{2}+\frac{a+c}{2}}{3}\right)^3=4.$$
The equality occurs for $z=0$, $x=2$ and $y=1$, which says that the answer is $4$.
