Inequality Proof

  • Thread starter rbzima
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  • #1
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Homework Statement



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


Homework Equations



None

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...
 

Answers and Replies

  • #2
VietDao29
Homework Helper
1,423
2

Homework Statement



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


Homework Equations



None

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...
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
 
  • #3
84
0
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

Wow, long night...
http://scienceblogs.com/insolence/facepalm.jpg [Broken]​
[/URL]
 
Last edited by a moderator:
  • #4
365
0
Just use (A-B)2=A2-2AB+B2.

Regards.
 
  • #5
446
1

Homework Statement



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


Homework Equations



None

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...

Refer to Cauchy–Schwarz inequality for more information (=
 

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