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anemone
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Let $x,\,y,\,z$ be non-negative reals such that $x^2+y^2+z^2+xyz=4$.
Prove that $0\le xy+yz+zx-xyz\le 2$.
Prove that $0\le xy+yz+zx-xyz\le 2$.
Kyle said:By Cauchy inequality, we have $4=x^2+y^2+z^2+xyz \ge 4\sqrt{x^3 y^3 x^3}$
so $xyz \le 1$ (1)
Then, $xy+yz+zx-xyz \le x^2 +y^2 +z^2 -1$ (since (1) and $x^2+y^2+z^2 \ge xy+yz+zy$) (I think you already know how to prove it)
Thus, $xy+yz+zx-xyz \le 4-xyz-1 \le 4-1-1 =2$
So the right part of the given inequality is proved. But I still have no idea how to prove the other part...:( Can anyone continue ?
Fallen Angel said:Hi,
A way for the other inequality.
From $xyz\leq 1$ you can assume $x\leq 1$
factor $xy+yz+xz-xyz=y(x+z-xz)+xz$ and now is trivial.
This inequality represents a mathematical statement that compares the values of 0, xy+yz+zx, and xyz, with the boundaries of 0 and 2. It is used to analyze the relationship between three variables in an equation.
This inequality can be proven using algebraic manipulation and mathematical properties such as the AM-GM inequality. By rearranging the terms and applying these properties, the inequality can be simplified and proven to hold true.
This inequality has implications in various fields of science, such as physics, chemistry, and biology. It can be used to analyze relationships between multiple variables in equations and can also be applied in optimization problems.
Yes, this inequality can be applied to real-world scenarios where three variables are involved and their values need to be compared. For example, in economics, this inequality can be used to analyze the relationship between costs, revenues, and profits.
Yes, the bounds of 0 and 2 have specific meanings in this inequality. The lower bound of 0 signifies that the values cannot be negative, while the upper bound of 2 represents a limit or a maximum value that the sum of the three variables can reach.