Proving a/b+b/a >= 2 using Mathematical Proof | Homework Help

[TheMathNoob]

Homework Statement

Let a,b be in the positive reals. Prove a/b+b/a is >=2

Homework Equations

The Attempt at a Solution

I have no idea. Maybe add the two ratios: (a^2+b^2)/a*b and then try to analyze separately the numerator and denominator?

 

[jbriggs444]

Those a's and b's in the denominator are pesky. Why not multiply through by ab and see what that gives you?

 

[TheMathNoob]

Those a's and b's in the denominator are pesky. Why not multiply through by ab and see what that gives you?

I feel like this is related to the law of cosines

 

[jbriggs444]

I feel like this is related to the law of cosines

Your powers of pattern recognition are good, but there is another formula involving a², b² and 2ab that is simpler yet.

 

[TheMathNoob]

Your powers of pattern recognition are good, but there is another formula involving a², b² and 2ab that is simpler yet.

oh hahahahahah (a-b)^2>=0
when a=b (a-b)^2=0
when a!=b (a-b)^2>0
so (a-b)^2>=0