For what x an y is y ( x + 3 ) 2 &gt; 2 x - y - 3 ?

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For what x an y is             y                        (          x          +          3          )                2              &amp;gt;          2              x        -        y        -        3            ?

szilincsifs · Accepted Answer

Step 1The pairs             {      x      ,      y      }       satisfying            y                        (          x          +          3          )                2              &amp;gt;          2              x        -        y        -        3            also satisfy       y          (      x      -      y      -      3      )        &amp;gt;    2                  (        x        +        3        )            2      and also satisfy       y          (      x      -      y      -      3      )        -    2                  (        x        +        3        )            2        &amp;gt;    0  Calling now             f              (        x        ,        y        )              =    y          (      x      -      y      -      3      )        -    2                  (        x        +        3        )            2      Step 2let us find the local minima/maxima: Solving the conditions      ∇          f              (        x        ,        y        )              =          0      →      or             {                                    12            -            4            x            +            y            =            0                                                -            3            +            x            -            2            y            =            0                              with solution             {      x      =      3      ,      y      =      0      }      for f(x, y) we have             ∇      2              f              (        x        ,        y        )              =          (                                    -            4                                1                                                1                                -            2                              )       with eigenvalues            {      -      3      -                        2                    ,      -      3      +                        2                    }       so f(x, y) has a maximum which is a global maximum, at             {      x      =      3      ,      y      =      0      }      .Evaluating             f              (        3        ,        0        )              =    0   we see that there is no pair             {      x      ,      y      }       which obeys strictly the condition.

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