values of a in inequalityIf x y + x y z 3 &lt; a ( x + 4 y + 4 z ) and x , y , z &gt; 0 then a iswith the help of am gm inequality x y ≤ ( x + y 2 ) and x y z 3 ≤ ( x + y + z 3 ) so x y + x y z 3 ≤ x + y 2 + x + y + z 3 = 5 x + 5 y + 2 z 6 want be able to go further, could some help me

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values of   a in inequalityIf       x    y    +            x      y      z        3    &amp;lt;  a  (  x  +  4  y  +  4  z  ) and   x  ,  y  ,  z  &amp;gt;  0 then   a iswith the help of am gm inequality            x      y        ≤          (                        x          +          y                2            )       and                     x        y        z            3        ≤          (                        x          +          y          +          z                3            )      so             x      y        +                  x        y        z            3        ≤                  x        +        y            2        +                  x        +        y        +        z            3        =                  5        x        +        5        y        +        2        z            6      want be able to go further, could some help me

Abdiel Nelson · Accepted Answer

We&#039;ll prove that  x  +  4  y  +  4  z  ≥  3      (          x      y        +                  x        y        z            3        )    .Indeed, by AM-GM  4  z  +  x  +  4  y  −  3      x    y    =  4  z  +  2  ⋅            x      +      4      y      −      3              x        y              2    ≥  ≥  3            4      z                        (                                    x              +              4              y              −              3                              x                y                                      2                    )                2              3    =  3            z                        (          x          +          4          y          −          3                      x            y                    )                2              3    .Thus, it remains to prove that            (      x      +      4      y      −      3              x        y            )        2    ≥  x  y  ,which is  (      x    −  2      y        )    2    (  x  +  4  y  −  2      x    y    )  ≥  0.The equality occurs for   x  =  4  y and   y  =  4  zId est,   a  &amp;gt;      1    3   are all values of   a, for which the inequality      x    y    +            x      y      z        3    &amp;lt;  a  (  x  +  4  y  +  4  z  )is true for all positives   x,   y and   z

values of a in inequality If sqrt(xy)+root[3]{xyz}<a(x+4y+4z) and x,y,z>0 then a is with the help of am gm inequality sqrt(xy)<=((x+y)/(2)) and root[3]{xyz}<=((x+y+z)/(3)) so sqrt{xy}+root[3]{xyz}<=(x+y)/(2)+(x+y+z)/(3)=(5x+5y+2z)/(6) want be able to go further, could some help me

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