Question

CMR: a²+b²+c²-ab-bc-ca>=0 với mọi a, b, c

Answer 1

Cần CM :\(a^2+b^2+c^2-ab-bc-ca\)>=0

<=>\(2\cdot a^2+2\cdot b^2+2\cdot c^2-2ab-2bc-2ca\)>=0(1)

ta có \(2a^2+2b^2+2c^2-2ab-2bc-2ca\)=\(\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)\)

=\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2>=0\) =>(1) (luôn đúng)

vậy suy ra đpcm

Dấu = khi a=b=c

Answer 2

Ta có ( a - b - c )² >= 0

= ( a-b )² - 2(a-b)c + c² >= 0

= a² - 2ab + b² - 2ac + 2bc + c² >= 0

= a² + b² + c² - 2 ( ab - bc + ac ) >=0 (dpcm)