Question

a,b,c aparțin lui R.
Demonstrați ca:
(a+b +c)^2≥3(ab+ac+bc)

Answer 1

[tex](a+b+c)^2\geq3(ab+ac+bc)\\ a^2+b^2+c^2+2ab+2ac+2bc\geq3ab+3ac+3bc\\ a^2+b^2+c^2-ab-ac-bc\geq0\\ 2a^2+2b^2+2c^2-2ab-2ac-2bc\geq 0\\ a^2-2ab+b^2+a^2-2ac+c^2+b^2-2bc+c^2\geq 0\\ (a-b)^2+(a-c)^2+(b-c)^2\geq 0[/tex]