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}

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