Prove that x+1/x ≥ 2 for all x>0

[mercury]

ok i have a really stupid problem ...but i can't seem to get it...would appreciate any help...
how do u prove that x+1/x >= 2 for all x>0...
is the above state ment true in the first place..i think it is...but can't seem to prove it...i keep coming to the fact that it holds for x>1 and equality holds for x=1 but can't seem to prove it...obviously for 0<x<1...

 

[HallsofIvy]

Since you are specifying that x> 0, you can multiply through the inequality by x without any problem:

You want to get x+ 1/x>= 2 and, multiplying by x, that is the same as x²+ 1>= 2x.

That, of course, is the same as x²- 2x+ 1>= 0 and you KNOW that's true.

Well, HOW do you know it's true? Also remember you have to ARRIVE at "x+ 1/x>= 2", not start there. Start with what you KNOW is true and work backwards.

 

[KLscilevothma]

Another method is to use the fact that A.M. > = G.M. as x > 0

Where AM stands for arithmatic mean and GM stands for Geometric mean.