Explanation

(c)
Let $y = \frac{x^{2} + x + 2}{x^{2} + x + 1}$
$\Rightarrow x^{2}(y - 1) + x(y - 1) + (y - 2) = 0,\ \forall\ x \in R$
Now, $D \geq 0\ \Rightarrow (y - 1)^{2} - 4(y - 1)(y - 2) \geq 0$
$\Rightarrow \ \ (y - 1)\{(y - 1) - 4(y - 2)\} \geq 0$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (y - 1)( - 3y + 7) \geq 0$
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$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \leq y \leq \frac{7}{3}$