Question

If $a,b,c\in R$ and $a^2+b^2+c^2=1,$ then $ab+bc+ca$ lies in the interval

• A. $[\frac{1}{2},2]$
• B. $[-1,2]$
• C. $[-\frac{1}{2},1]$
• D. $[-1,\frac{1}{2}]$

Answer 1

The correct option is C $[-\frac{1}{2},1]$

$(a+b+c{)}^2\ge 0\Rightarrow 1+2(ab+bc+ca)\ge 0\Rightarrow ab+bc+ca\ge -\frac{1}{2}\cdots (1)$
And we know that
$(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ge 0\Rightarrow 2-2(ab+bc+ca)\ge 0\Rightarrow ab+bc+ca\le 1\cdots (2)$

From $\left(1\right)$ and $\left(2\right)$,
$ab+bc+ca\in [-\frac{1}{2},1]$