Given the positive numbers a , b , c. Prove that a a 2 + 1 + b b 2 + 1 + c c 2 + 1 ≤ 3 2 By Cauchy Schwarz, we have that ( a 2 + 1 ) ( 1 + 3 ) ≥ ( a + 3 ) 2 → a a 2 + 1 ≤ 2 a a + 3 I need a new method

Question

Given the positive numbers   a  ,  b  ,  c. Prove that       a                  a        2            +      1        +      b                  b        2            +      1        +      c                  c        2            +      1        ≤      3    2  By Cauchy Schwarz, we have that  (      a    2    +  1  )  (  1  +  3  )  ≥            (      a      +              3            )        2    →      a                  a        2            +      1        ≤            2      a              a      +              3            I need a new method

megagoalai · Accepted Answer

Note that   f  (  x  )  =      x                  x        2            +      1       is a concave function for   x  &amp;gt;  0. This follows as      f    ″    (  x  )  =  −                    3        x                    (                  x          2                +        1                  )                                    5              2                                            &amp;lt;  0Now use Jensen&#039;s inequality. Note that  f  (  a  )  +  f  (  b  )  +  f  (  c  )  ≤  3  f      (                  a        +        b        +        c            3        )    =  3  f      (                  3            3        )    =      3    2  As   a  +  b  +  c  =      3  The maximum is achieved when   a  =  b  =  c. We are done!

Ignacio Riggs · Accepted Answer

Another way.Since   a  b  +  a  c  +  b  c  ≤      1    3    (  a  +  b  +  c      )    2    ≤  1, by AM-GM we obtain:      ∑          c      y      c            a          1      +              a        2              ≤      ∑          c      y      c            a          a      b      +      a      c      +      b      c      +              a        2              =      ∑          c      y      c            a          (      a      +      b      )      (      a      +      c      )        ≤  ≤      1    2        ∑          c      y      c            (          a              a        +        b              +          a              a        +        c              )    =      1    2        ∑          c      y      c            (          a              a        +        b              +          b              a        +        b              )    =      3    2    .Done!

Given the positive numbers a,b,c. Prove that (a)/(sqrt(a^2+1))+(b)/(sqrt(b^2+1))+(c)/(sqrt(c^2+1))<=(3)/(2)

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