Home > Uncategorized > A test, also an interesting problem

## A test, also an interesting problem

This post is used for testing the appearance of $\LaTeX$.
Problem:

Given nonnegative real numbers $x$, $y$ and $z$ which satisfy
$x+y+z=1$. Your task is to find the maximum of $S=x^2yz+xy^{2}z$.

Solution:

According to GM-AM inequality, for any nonnegative real numbers $x$,
$y$ we have $xy<=\frac{(x+y)^2}{4}$. Substitute the corresponding
item in S, we get

$S=x^2yz+xy^2z=(x+y)\cdot xyz=(1-z) \cdot xyz<=(1-z) \cdot \frac{(x+y)^2}{4} \cdot z=\frac{1}{3} \cdot (1-z)\cdot \frac{(1-z)^2}{4} \cdot 3z$

Using GM-AM inequality again, we get

$S<=\frac{1}{3} \cdot \frac{(1-z+1-z+1-z+3z)^4}{256\cdot4}=\frac{27}{1024}$