Proofs

[kevi555]

Given two numbers a and b (a, b E R), use indirect proof to prove that a^2 +b^2 is greater and equal to 2ab. Any ideas? Any help would be greatly appreciated.

 

Kev

[Callipygous]

Given two numbers a and b (a' date=' b E R), use indirect proof to prove that a^2 +b^2 is greater and equal to 2ab. Any ideas? Any help would be greatly appreciated.

 

Kev[/quote']

 

i might be able to help, but its been a while since algebra. whats (a,b E R)?

[kevi555]

(a, b E R) just means that a and b are elements of the real numbers..

[Callipygous]

yeah... im out. i knew there was a reason i hated this stuff...

 

sorry. : P

 

good luck

[TD]

Given two numbers a and b (a' date=' b E R), use indirect proof to prove that a^2 +b^2 is greater and equal to 2ab. Any ideas? Any help would be greatly appreciated.

 

Kev[/quote']

We know that

 

[math]\left( {a - b} \right)^2 = a^2 - 2ab + b^2 [/math]

 

But since a square of a real number is never negative, we have

 

[math]\left( {a - b} \right)^2 \geqslant 0 \Leftrightarrow a^2 - 2ab + b^2 \geqslant 0 \Leftrightarrow a^2 + b^2 \geqslant 2ab[/math]