inequality

[kanali]

Prove the following inequality:

$\displaystyle xyz\geq x+y+z-2$ for $\displaystyle x\geq 1,y\geq 1,z\geq 1$.


i can only think of squaring the equation but would that help??

 

[Deleted member 4993]

Prove the following inequality:

$\displaystyle xyz\geq x+y+z-2$ for $\displaystyle x\geq 1,y\geq 1,z\geq 1$.


i can only think of squaring the equation but would that help??

I have not totally thought it through - but I think following identity might help:

$\displaystyle a^3 \ \ + \ \ b^3 \ \ + \ \ c^3 \ \ - \ \ 3abc \ \ = \ \ (a \ \ + \ \ b \ \ + \ \ c)\cdot (a^2 \ \ + \ \ b^2 \ \ + \ \ c^2 \ \ - \ \ ab \ \ - \ \ bc \ \ - \ \ ca)$