If \( f: D \rightarrow R, f(x)=\frac{x^{2}+b x+c}{x^{2}+b_{1} x+c_{1}} \), where \( \alpha, \bet...

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If \( f: D \rightarrow R, f(x)=\frac{x^{2}+b x+c}{x^{2}+b_{1} x+c_{1}} \), where \( \alpha, \beta \) are the roots of the equation \( x^{2}+b x+c=0 \) and \( \alpha_{1}, \beta_{1} \) are the roots of \( x^{2}+b_{1} x+c_{1}=0 \). Now, answer the following questions for \( f(x) \). A combination of graphical and analytical approach may be helpful in solving these problems. (If \( \alpha_{1} \) and \( \beta_{1} \) are real then \( f(x) \) has vertical asymptote at \( \left.x=\alpha_{1}, \beta_{1}\right) \). Then,
If \( \alpha_{1}\beta_{1}\alpha\beta \), then
(a) \( f(x) \) has a maxima in \( \left[\alpha_{1}, \beta_{1}\right] \) and a minima is \( [\alpha, \beta] \)
(b) \( f(x) \) has a minima in \( \left(\alpha_{1}, \beta_{1}\right) \) and a maxima in \( (\alpha, \beta) \)
(c) \( f^{\prime}(x)0 \) where ever defined
(d) \( f^{\prime}(x)0 \) where ever defined
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