Question

Cho a,b,c thuộc R. Cm: (a+b+c/3 )^2 lớn hơn hoặc bằng ab+bc+ca

Answer 1

(a+b+c/3)²= a²+b²+(c/3)²+2ab+2/3ac+2/3bc

* a²+b²+(c/3)² \(\ge\)0

=> a²+b²+(c/3)²+2ab+2/3ac+2/3bc\(\ge\)2ab+2/3ac+2/3bc

mà 2ab+2/3ac+2/3bc\(\ge\)ab+bc+ca

=> a²+b²+(c/3)²+2ab+2/3ac+2/3bc\(\ge\)ab+bc+ca

=> (a+b+c/3)²\(\ge\)ab+bc+ca

Answer 2

trả lời:

(a+b+c/3)2= a2+b2+(c/3)2+2ab+2/3ac+2/3bc

* a2+b2+(c/3)2 \ge≥0

=> a2+b2+(c/3)2+2ab+2/3ac+2/3bc\ge≥2ab+2/3ac+2/3bc

mà 2ab+2/3ac+2/3bc\ge≥ab+bc+ca

=> a2+b2+(c/3)2+2ab+2/3ac+2/3bc\ge≥ab+bc+ca

=> (a+b+c/3)2\ge≥ab+bc+ca