Evaluation Of Maximum Value Without Calculus

in general

if $x,y,z\ge 0$,and such $x+y+z=3$, then we have $$x^ky+y^kz+z^kx\le\max{\{3,\dfrac{3^{k+1}k^k}{(k+1)^{k+1}}\}}$$

For $k=2$ I have nice methods

with out loss of let $x=\max{(x,y,z)}$

then we use Benoulli inequality,we have $$(1+\dfrac{z}{x})^2\ge 1+\dfrac{2z}{x}$$ so $$(x+z)^2y=x^2y(1+\frac zx)^2\ge x^2y(1+\dfrac{2z}{x})=x^2y+xyz+xyz\ge x^2y+y\cdot yz+xz^2$$ so $$x^2y+y^2z+z^2x\le (x+z)^2y=2^2\left(\dfrac{x+z}{2}\right)^2\cdot y\le 2^2\left(\dfrac{x+z+y}{3}\right)^3=4$$