A) \[(-\infty ,3]\]
B) \[(-\infty ,\,\infty )\]
C) \[[3,\infty )\]
D) \[\left[ \frac{1}{3},3 \right]\]
E) \[\left( -\infty ,\frac{1}{3} \right)\cup (3,\infty )\]
Correct Answer: D
Solution :
Let \[y=\frac{{{x}^{2}}-x+1}{{{x}^{2}}+x+1}\] \[\Rightarrow \] \[{{x}^{2}}(y-1)+4(y+1)+(y-1)=0\] Now, \[D\ge 0\] \[\Rightarrow \] \[{{(y+1)}^{2}}-4{{(y-1)}^{2}}\ge 0\] \[\Rightarrow \] \[-3{{y}^{2}}-3+10y\ge 0\] \[\Rightarrow \] \[3{{y}^{2}}-10y+3\le 0\] \[\Rightarrow \] \[y=\frac{10\pm \sqrt{64}}{6}\le 0\] \[\left( y-\frac{1}{3} \right)(y-3)\le 0\] \[\Rightarrow \] \[y\in \left[ \frac{1}{3},3 \right]\]
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