Given a + b + c + d = 4 To find Minimum value of a b 3 + 4 + b c 3 + 4 + c d 3 + 4 + d a 3 + 4 Given a + b + c + d = 4 where a , b , c , d ∈ R + find Minimum value of S = a b 3 + 4 + b c 3 + 4 + c d 3 + 4 + d a 3 + 4 I have no clue to start...any hint?

Question

Given   a  +  b  +  c  +  d  =  4 To find Minimum value of       a                  b        3            +      4        +      b                  c        3            +      4        +      c                  d        3            +      4        +      d                  a        3            +      4      Given   a  +  b  +  c  +  d  =  4 where   a  ,  b  ,  c  ,  d  ∈            R              +            find Minimum value of  S  =      a                  b        3            +      4        +      b                  c        3            +      4        +      c                  d        3            +      4        +      d                  a        3            +      4      I have no clue to start...any hint?

hitturn35 · Accepted Answer

By AM-GM      ∑          c      y      c            a                  b        3            +      4        =            a      +      b      +      c      +      d        4    −      ∑          c      y      c            (          a      4        −          a                        b          3                +        4              )    =  =  1  −      ∑          c      y      c                          b        3            a              4              (                              b            3                    2                +                              b            3                    2                +        4        )              ≥  1  −                    b        3            a              4      ⋅      3                                                  (                                                b                  3                                2                            )                        2                    ⋅          4                3              =  =  1  −      ∑          c      y      c                          b        3            a              12              b        2              =  1  −      1    12    (  a  b  +  b  c  +  c  d  +  d  a  )  =  =  1  −      1    12    (  a  +  c  )  (  b  +  d  )  ≥  1  −                    (                              a            +            b            +            c            +            d                    2                )            2        12    =  1  −      1    3    =      2    3    .For   (  a  ,  b  ,  c  ,  d  )  →  (  2  ,  2  ,  0  ,  0  ) we see that   S  →      2    3  ,which says that       2    3   is infimumand the minimum does not exist.Done!

Given a+b+c+d=4 where a,b,c,d in RR^+ find Minimum value of S=(a)/(b^3+4)+(b)/(c^3+4)+(c)/(d^3+4)+(d)/(a^3+4) I have no clue to start...any hint?

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