Inequality Proof

tom_smith

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Mar 24, 2021
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I've been stuck on this question for a while now, a^2+b^2+c^2≥ ab+bc+ca

I get the basic (a-b)^2 ≥ 0,
a^2+b^2≥2ab, but I can't figure out how to get this one out.

Thanks for any help.
 
I've been stuck on this question for a while now, a^2+b^2+c^2≥ ab+bc+ca


Here is a possible hint:

2(a2+b2+c2)  2(ab+bc+ca)\displaystyle 2(a^2 + b^2 + c^2) \ \ge \ 2(ab + bc + ca)


Here is a more advanced hint:
a2            b2\displaystyle a^2 \ \ \ \ \ \ \ \ \ \ \ \ b^2
b2            c2\displaystyle b^2 \ \ \ \ \ \ \ \ \ \ \ \ c^2
c2            a2  2ab+2bc+2ca\displaystyle c^2 \ \ \ \ \ \ \ \ \ \ \ \ a^2 \ \ge \ 2ab + 2bc + 2ca
 
[MATH]a^2+b^2 \ge 2ab[/MATH]
[MATH]b^2+c^2 \ge 2bc[/MATH]
[MATH]a^2+c^2 \ge 2ac[/MATH]
sum of the left sides [MATH]\ge[/MATH] sum of the right sides
 
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