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Inequality Proof

  1. Aug 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that [tex]\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)[/tex]


    2. Relevant equations

    None

    3. The attempt at a solution

    I've broken the LHS down to the following:

    [tex]\left(ab\right)^{2}+2abcd+\left(cd\right)^{2} [/tex]

    The RHS:

    [tex]\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}[/tex]

    So, ultimately... it works out that I need to show [tex]2abcd \leq \left(ad\right)^{2} + \left(bc\right)^{2}[/tex]

    This is where I'm getting stuck... Any suggestions...
     
  2. jcsd
  3. Aug 25, 2009 #2

    VietDao29

    User Avatar
    Homework Helper

    Yup, so far so good, now subtract 2abcd from both sides, and you'll get:

    [tex](ad) ^ 2 - 2(ad)(bc) + (bc) ^ 2 \geq 0[/tex]

    Does the LHS of this inequality remind you of something? :surprised
     
  4. Aug 25, 2009 #3

    Wow, long night...
    http://scienceblogs.com/insolence/facepalm.jpg [Broken]​
    [/URL]
     
    Last edited by a moderator: May 4, 2017
  5. Aug 26, 2009 #4
    Just use (A-B)2=A2-2AB+B2.

    Regards.
     
  6. Aug 27, 2009 #5

    Refer to Cauchy–Schwarz inequality for more information (=
     
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