In resticted domain , Applying the Cauchy-Schwarz's inequalityI see this problem a few days ago. a , b , c ∈ [ α , β ]prove that 9 ≤ ( a + b + c ) ( 1 a + 1 b + 1 c ) ≤ ( 2 α + β ) ( α + 2 β ) α β Left inequality is easy by Cauchy-Schwart's inequality. But Right inequality is difficult to me. How can I approach the Right inequlity?EDIT : Sorry, I forgot the condition : α , β &gt; 0

Question

In resticted domain , Applying the Cauchy-Schwarz&#039;s inequalityI see this problem a few days ago.  a  ,  b  ,  c  ∈  [  α  ,  β  ]prove that  9  ≤  (  a  +  b  +  c  )            (            1    a    +      1    b    +      1    c              )        ≤            (      2      α      +      β      )      (      α      +      2      β      )              α      β      Left inequality is easy by Cauchy-Schwart&#039;s inequality. But Right inequality is difficult to me. How can I approach the Right inequlity?EDIT : Sorry, I forgot the condition :   α  ,  β  &amp;gt;  0

Dana Simmons · Accepted Answer

I hope you mean that   α  &amp;gt;  0Let   f  (  a  ,  b  ,  c  )  =  (  a  +  b  +  c  )      (          1      a        +          1      b        +          1      c        )  Thus,   f is a convex function of   a, of   b and of   c (for example,             ∂      f              ∂              a        2              =            2      (      b      +      c      )              a      3        &amp;gt;  0), which says that      max          {      a      ,      b      ,      c      }      ⊂      [      α      ,      β      ]        f  =      max          {      a      ,      b      ,      c      }      ⊂      {      α      ,      β      }        f  =  f  (  α  ,  α  ,  β  )  =            (      2      α      +      β      )      (      α      +      2      β      )              α      β      because   f  (  α  ,  α  ,  α  )  =  f  (  β  ,  β  ,  β  )  =  9 and by AM-GM            (      2      α      +      β      )      (      α      +      2      β      )              α      β        =  2      (          α      β        +          β      α        )    +  5  ≥  4  +  5  =  9.

In resticted domain , Applying the Cauchy-Schwarz's inequality I see this problem a few days ago. a, b, c in [alpha , beta] prove that 9 <= (a+b+c)(1/a +1/b +1/c)<= ((2 alpha + beta)(alpha + 2 beta ))/( alpha beta)

Answered question

Answer & Explanation

New Questions in Pre-Algebra