The exhaustive set of values of b such that ${I n (x^{2} - 2 x + 2)}^{2} + b I n (x^{2} - 2 x + 2) + 1 > 0 \forall x > 1$ is

(- 3, \infty)

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(- 1, \infty)

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(- 2, \infty)

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(2, \infty)

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Solution

The correct option is C $(- 2, \infty)$
Let $t = I n (x^{2} - 2 x + 2) = I n [(x - 1)^{2} + 1]$
Then t > 0 $\forall$ x > 1. So we need to find b such that $t^{2}$ + bt + 1 > 0 $\forall$ t > 0
$\Rightarrow b > - (t + \frac{1}{t}) \Rightarrow b > - 2$

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