I want to prove that f ( x , y ) = ( x − 2 y ) 4 + 64 x y + 16 ≥ 0

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I want to prove that  f  (  x  ,  y  )  =  (  x  −  2  y      )    4    +  64  x  y  +  16  ≥  0

laure6237ma · Accepted Answer

You can complete the square.  f  (  x  ,  y  )  =  (  x  −  2  y      )    4    +  64  x  y  +  16Note that   4  x  y  =      x    2    +  4      y    2    −  (  x  −  2  y      )    2  So   f  (  x  ,  y  )  =  (  x  −  2  y      )    4    +  8  [      x    2    +  4      y    2    −  (  x  −  2  y      )    2    ]  +  32  x  y  +  16  =  (  x  −  2  y      )    4    −  8  (  x  −  2  y      )    2    +  16  +  8  (      x    2    +  4      y    2    +  4  x  y  )  =  [  (  x  −  2  y      )    2    −  4      ]    2    +  8  (  x  +  2  y      )    2    ≥  0

landdenaw · Accepted Answer

Write   t  =  x  −  2  y so   x  =  t  +  2  y and now we have                    f        (        t        ,        y        )                            =                  t          4                +        64        y        (        t        +        2        y        )        +        16                                          =        128                  y          2                +        64        y        t        +        (                  t          4                +        16        )            This is quadratic function on y. It discriminant is                    D                            =                  64          2                          t          2                −        4        ⋅        128        (                  t          4                +        16        )                                          =        −        128        ⋅        4        (                  t          4                −        8                  t          2                +        16        )                                          =        −        128        ⋅        4        (                  t          2                −        4                  )          2                                                  ≤        0

I want to prove that f ( x , y ) = ( x − 2 y )

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