1. Apr 20, 2016 #1

   TheMathNoob

   

   1. The problem statement, all variables and given/known data
   Let a,b be in the positive reals. Prove a/b+b/a is >=2
   2. Relevant equations
   
   
   3. 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?

    

   TheMathNoob, Apr 20, 2016
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3. Apr 20, 2016 #2

   jbriggs444

   

   Science Advisor

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

    

   jbriggs444, Apr 20, 2016
4. Apr 20, 2016 #3

   TheMathNoob

   

   I feel like this is related to the law of cosines

    

   TheMathNoob, Apr 20, 2016
5. Apr 20, 2016 #4

   jbriggs444

   

   Science Advisor

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

    

   jbriggs444, Apr 20, 2016
6. Apr 20, 2016 #5

   TheMathNoob

   

   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

    

   TheMathNoob, Apr 20, 2016

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