Schwarz inequality

  • #1
890
39

Homework Statement


For x,y,z ## \in \mathbb {R^+} ##, prove that
## \sqrt {x (3 x +y) } + \sqrt {y (3y +z) } + \sqrt {z(3z +x)} \leq ~ 2(x +y+ z) ##


Homework Equations


upload_2018-8-16_11-49-16.png


The Attempt at a Solution


I don't know which inequality among the above two has to be applied.
I am trying to solve it by inspection. I don't know the standard approach to solve it.
 

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Answers and Replies

  • #2
35,328
11,637
If in doubt, write the Schwarz inequality with components of general vectors v,w, expand the smaller side, then see if you can assign the components to get these square roots at a suitable spot in the equation.
 
  • #3
890
39
If in doubt, write the Schwarz inequality with components of general vectors v,w, expand the smaller side, then see if you can assign the components to get these square roots at a suitable spot in the equation.

I did it. Thanks a lot for guiding me.
 

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