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Consider the function \( f:(-\infty, \infty) \rightarrow(-\infty, \infty) \) defined by
\( \mathrm{P} \) \( f(x)=\frac{x^{2}-a x+1}{x^{2}+a x+1} ; 0a2 \).
\( (2008,12 \mathrm{M}) \)
W
Let \( g(x)=\int_{0}^{e^{x}} \frac{f^{\prime}(t)}{1+t^{2}} d t \). Which of the following is true?
(a) \( g^{\prime}(x) \) is positive on \( (-\infty, 0) \) and negative on \( (0, \infty) \)
(b) \( g^{\prime}(x) \) is negative on \( (-\infty, 0) \) and positive on \( (0, \infty) \)
(c) \( g^{\prime}(x) \) changes sign on both \( (-\infty, 0) \) and \( (0, \infty) \)
(d) \( g^{\prime}(x) \) does not change \( \operatorname{sign}(-\infty, \infty) \)
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