What is the minimum value of 8 x 3 + 36 x + 54 / x + 27 / x 3 for positive real numbers x?

Question

What is the minimum value of   8      x    3    +  36  x  +  54      /    x  +  27      /        x    3   for positive real numbers x?

Haleigh Vega · Accepted Answer

Step 1For numbers       a    1    ,  …  ,      a    n    &amp;gt;  0 , the AM-GM inequality is                               a          1                +                  a          2                +        ⋯        +                  a          n                    n        ≥                    a        1                    a        2            ⋯              a        n              n   .Equality occurs iff       a    1    =      a    2    =  ⋯  =      a    n   .The trick here is to apply AM-GM to the right number of terms at a time.Since x is positive, the 4 terms   8      x    3   ,   36  x,             54      x       , and             27              x        3             are all positive.Hence,                     8                  x          3                +        36        x        +                              54            x                          +                              27                          x              3                                          4        ≥            8              x        3            ⋅      36      x      ⋅                        54          x                    ⋅                        27                      x            3                                4  For equality to occur, you need   8      x    3    =  36  x  =            54      x        =            27              x        3             . Unfortunately, there is no such x for which all four are equal.Instead, let&#039;s apply AM-GM to two terms at a time:                    8                  x          3                +                              27                          x              3                                          2        ≥      8          x      3        ⋅                  27                  x          3                      =      8    ⋅    27    =  6      6    ⇝  8      x    3    +            27              x        3              ≥  12      6                      36        x        +                              54            x                              2        ≥      36    x    ⋅                  54        x              =      36    ⋅    54    =  18      6    ⇝  36  x  +            54      x        ≥  36      6  Now, add the two together to get   8      x    3    +  36  x  +            54      x        +            27              x        3              ≥  48      6   .Finally, equality occurs iff   8      x    3    =            27              x        3             and   36  x  =            54      x       . There is in fact a positive number x for which both are satisfied (specifically   x  =                    3        2             ).Alternatively, if you notice that   8      x    3    +  36  x  +            54      x        +            27              x        3              =            (      2      x      +                        3          x                    )        3   as Alexey Burdin pointed out in the comments, then you can just apply AM-GM to   2  x  +            3      x       .

Simone Werner · Accepted Answer

Step 1To use AM-GM and find minimum, you need to have equality possible. This can happen iff all the terms you use can be made equal for some value of x. Let&#039;s try finding this first.  8      x    3    =  27      /        x    3      ⟹    x  =            3      2       . At this value, the terms are   6      6    ,  18      6    ,  18      6    ,  6      6   respectively, so this suggests we use AM-GM in this form:  8      x    3    +  12  x  +  12  x  +  12  x  +  18      /    x  +  18      /    x  +  18      /    x  +  27      /        x    3      ≥  8      8.12.12.12.18.18.18.27    8    =  48      6  and have guarantee that value is in fact attained for the said   x  =            3      2       .

What is the minimum value of 8 x 3 </msup> + 36 x + 54 <mrow

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