Question

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question 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 $x=({\alpha }_1,{\beta }_1$). Then if ${\alpha }_1<\alpha <{\beta }_1<\beta$, then

• A. f(x) is increasing in ${\alpha }_1,{\beta }_1$
• B. f(x) is decreasing in $\alpha ,\beta$
• C. f(x) is decreasing in ${\beta }_1,\beta$
• D. f(x) is decreasing in $(-\infty ,\alpha )$

Answer 1

The correct option is A

f(x) is increasing in ${\alpha }_1,{\beta }_1$

Graph of f(x) is shown

Clearly, f(x) is increasing in ${\alpha }_1,{\beta }_1$