A) \[\left[ -\frac{1}{2},\infty \right)\]
B) \[[0,\infty )\]
C) \[\left[ -\frac{1}{2},1 \right]\]
D) \[[1,\infty )\]
Correct Answer: A
Solution :
Given, \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\] \[\because \] \[{{(a+b+c)}^{2}}\ge 0\] \[\Rightarrow \] \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca\ge 0\] \[\Rightarrow \] \[2(ab+bc+ca)\ge -1\] \[\Rightarrow \] \[ab+bc+ca\ge -\frac{1}{2}\] \[\therefore \] Range is \[\left[ -\frac{1}{2},\infty \right)\]
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