1. ## Inequality

(x+y)*(x+z)*(y+z)>=8xyz

How to show this, I suppose somehow with the AM–GM inequality, but don't know how.

Thank you!

2. ## Re: Inequality

Originally Posted by gotmejerry
(x+y)*(x+z)*(y+z)>=8xyz

How to show this, I suppose somehow with the AM–GM inequality, but don't know how.

Thank you!

$\displaystyle 1) x+y \geq 2\sqrt {xy}$

$\displaystyle 2) x+z \geq 2\sqrt {xz}$

$\displaystyle 3) y+z \geq 2\sqrt {yz}$

Now multiply left hand sides and right hand sides of these three equations .

Thank you!

4. ## Re: Inequality

you are welcome

5. ## Re: Inequality

Just nitpicking...

For x=1, y=-1, z=-1, you get:

$\displaystyle (1-1)(1-1)(-1-1) \ge 8 \cdot 1 \cdot -1 \cdot -1$

$\displaystyle \Rightarrow 0 \ge 8$

6. ## Re: Inequality

Originally Posted by gotmejerry
(x+y)*(x+z)*(y+z)>=8xyz
How to show this, I suppose somehow with the AM–GM inequality, but don't know how.
I assume that $\displaystyle x,~y,~\&~z$ are non-negative.
We have $\displaystyle x^2+y^2\ge 2xy.$ so $\displaystyle x^2z+y^2z\ge 2xyz.$.

Expand $\displaystyle (x+y)(x+z)(y+z)$.
Do you see what to do next?

7. ## Re: Inequality

Sorry, I forgot that x,y,z are postives