prove that..?

Question

Prove that (cuberoot of xyz)*(1/x+1/y+1/z)is greater than or equal to
3 where x,y and z are positive real numbers.

Answer 1

ahh, no answers yet. I worked on it a little, but didn't have too much time...here is where I got to (all I did was some basic algebra...nothing heavy duty):

(xy + xz + yz) / (xyz)^(2/3) >= 3

If you can show that this ratio is always bigger than or equal to 3, then you will have it. I suggest that you might have to look at two or three cases.

xyz >1
xyz=1
xyz<1

But I could be way off on that. It's been some time since I took my Numerical Analysis class, so some of this stuff has slipped through the cracks.

I will keep thinking about this...hope that helped up at least a little bit

Answer 2

1?