Question

cho a,b,c  thỏa mãn a^2+b^2+c^2=1     CMR  -1/2<= ab+bc+ca<=1

Answer 1

\(a^2+b^2+c^2=1\)

\(\Leftrightarrow a^2+b^2+c^2+2ab+2bc+2ca=1+2\left(ab+bc+ca\right)\)

\(\Leftrightarrow\left(a+b+c\right)^2=1+2\left(ab+bc+ca\right)\)

\(\Rightarrow1+2\left(ab+bc+ca\right)\ge0\)

\(\Rightarrow ab+bc+ca\ge-\dfrac{1}{2}\)

Lại có:

\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)

\(\Leftrightarrow2ab+2bc+2ca\le2a^2+2b^2+2c^2\)

\(\Leftrightarrow ab+bc+ca\le a^2+b^2+c^2\)

\(\Leftrightarrow ab+bc+ca\le1\)