Prove AM-GM Inequality: a,b,c ≥ 0 and a+b+c=3

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The discussion centers on proving the AM-GM inequality for non-negative variables a, b, and c, given that a + b + c = 3. The key inequality to prove is a² + b² + c² + ab + ac + bc ≥ 6. The approach involves rewriting the left-hand side using the square of the sum of the variables, leading to the conclusion that ab + ac + bc ≤ 3. This establishes the necessary condition for the AM-GM inequality under the specified constraints.

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klajv
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Can't figure this out and hope to get some help, TIA!

a,b,c >= 0 and a+b+c=3
Prove that a²+b²+c²+ab+bc+ca >= 6
 
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Write the left hand side of the inequality differently, using the expression for (a+b+c)^2.

You will arrive upon the inequality ab+bc+ac <= 3 = a+b+c. Now, can you prove ab+bc+ac <= (a+b+c)(a+b+c)/3 ?
 
disregardthat <---

Yes, thanks a lot
 

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