if $x+y+z=1$ then find the maximum of the form 
Let $x>0$, $y>0$ and $z>0$ such that $x+y+z=1$. Find the maximal value of
  $$x\sqrt{y}+y\sqrt{z}$$

I think $x=y=\dfrac{4}{9},z=\dfrac{1}{9}$ then the maximum $\dfrac{4}{9}$,but how to use AM-GM prove it?

Without AM-GM, the Lagrange multiplier method will do:
$$L=x\sqrt{y}+y\sqrt{z}+t(1-x-y-z).$$
$$\begin{cases} L_x=\sqrt{y}-t=0 \\ L_y=\frac{x}{2\sqrt{y}}+\sqrt{z}-t=0 \\ L_z=\frac{y}{2\sqrt{z}}-t=0 \\ L_t=1-(x+y+z)=0 \end{cases} \Rightarrow x=y=\frac{4}{9}, z=\frac{1}{9}.$$
Note: To solve the system of equations, square $(1)$ divide it by $(3)$ to find $t=2\sqrt{z}$. Then substitute this into $(1)$ to find $y=4z$. Then substitute these into $(2)$ to find $x=4z$. And finally substitute all into $(4)$ to find $z=\frac{1}{9}$, hence the rest.
