Prove Inequality: (ab+cd)^2 ≤ (a^2+c^2)(b^2+d^2)

[rbzima]

Homework Statement

Prove that $$\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)$$

Homework Equations

None

The Attempt at a Solution

I've broken the LHS down to the following:

$$\left(ab\right)^{2}+2abcd+\left(cd\right)^{2}$$

The RHS:

$$\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}$$

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

This is where I'm getting stuck... Any suggestions...

 

[VietDao29]

Homework Statement

Prove that $$\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)$$

Homework Equations

None

The Attempt at a Solution

I've broken the LHS down to the following:

$$\left(ab\right)^{2}+2abcd+\left(cd\right)^{2}$$

The RHS:

$$\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}$$

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

This is where I'm getting stuck... Any suggestions...

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

$$(ad) ^ 2 - 2(ad)(bc) + (bc) ^ 2 \geq 0$$

Does the LHS of this inequality remind you of something?

 

[rbzima]

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

$$(ad) ^ 2 - 2(ad)(bc) + (bc) ^ 2 \geq 0$$

Does the LHS of this inequality remind you of something?

Wow, long night...

http://scienceblogs.com/insolence/facepalm.jpg​

[/URL]

 

[Дьявол]

Just use (A-B)²=A²-2AB+B².

Regards.

 

[icystrike]

Homework Statement

Prove that $$\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)$$

Homework Equations

None

The Attempt at a Solution

I've broken the LHS down to the following:

$$\left(ab\right)^{2}+2abcd+\left(cd\right)^{2}$$

The RHS:

$$\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}$$

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

This is where I'm getting stuck... Any suggestions...

Refer to Cauchy–Schwarz inequality for more information (=