How to prove that x y + y x ≥ x + y I am trying to prove that x y + y x ≥ x + y I tried some manipulations, like multiplications in x or y or using x = x x , but I'm still stuck with that.What am I missing?

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How to prove that       x          y        +      y          x        ≥      x    +      y  I am trying to prove that      x          y        +      y          x        ≥      x    +      y  I tried some manipulations, like multiplications in       x   or       y   or using   x  =      x        x  , but I&#039;m still stuck with that.What am I missing?

Easton Chambers · Accepted Answer

Let   a  &amp;gt;  0,   b  &amp;gt;  0, we have that                                          a            2                    b                +                              b            2                    a                ≥        a        +        b                                    ⟺                                                    a              3                        +                          b              3                                            a            b                          ≥        a        +        b                                                  ⟺                          a          2                −        a        b        +                  b          2                ≥        a        b                                                  ⟺                (        a        −        b                  )          2                ≥        0

stud4kuj5bwn · Accepted Answer

Diviging by       y   we get,      x    y    +            y              x        ≥            x      y        +  1which can be written      t    2    +      1    t    ≥  t  +  1or      t    3    −      t    2    −  t  +  1  =  (  t  −  1      )    2    (  t  +  1  )  ≥  0for   t  &amp;gt;  0. Equality occurs with   t  =  1, i.e.   x  =  y

How to prove that x <msqrt> y </msqrt> </mfrac> + y <

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