show that c 2 a + a 2 c + 16 a c ≥ 9 ( a + c )Let a , c &gt; 0 show that c 2 a + a 2 c + 16 a c ≥ 9 ( a + c )It seem AM-GM inequality.How?

Question

show that             c      2        a    +            a      2        c    +  16      a    c    ≥  9  (  a  +  c  )Let   a  ,  c  &amp;gt;  0 show that                    c        2            a        +                    a        2            c        +  16      a    c    ≥  9  (  a  +  c  )It seem AM-GM inequality.How?

Aiden Norman · Accepted Answer

Yes, we can use AM-GM!Let             a      c        +            c      a        =  uHence, we need to prove that       u    3    −  3  u  +  16  ≥  9  u or      u    3    +  8  +  8  ≥  12  u, which is AM-GM.

Emmy Knox · Accepted Answer

Certainly an Overkill            c      2        a    +            a      2        c    +  16      a    c    ≥  9  (  a  +  c  )  ⟺      (                  c        2            a        −    a    )    +      (                  a        2            c        −    c    )    −  8  (      c    −      a        )    2    ≥  0  ⟺  (      c    −      a        )    2        [    (          c        +          a              )      2        (    c    +    a    )    −    8    a    c    ]    ≥  0  ⟺  (      c    +      a        )    2    (  c  +  a  )  −  8  a  c  ≥  0                    (*)                    ⟺        (        a        +        c                  )          2                +        2        (        a        +        c        )                  a          c                −        8        a        c        ≥        0            By AM-GM Inequality                    (1)                    (        a        +        c                  )          2                ≥        4        a        c                                (2)                    2        (        a        +        c        )                  a          c                ≥        4        a        c            Thus, we see that (∗) follows from (1) and (2)

show that c 2 </msup> a </mfrac> + a

Answered question

Answer & Explanation

New Questions in Pre-Algebra