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 $x=({\alpha }_1,{\beta }_1$).
If ${\alpha }_1<{\beta }_1<\alpha <\beta$, then

• A. f(x) has a maxima in $[{\alpha }_1,{\beta }_1]$ and a minima is $[\alpha ,\beta ]$
• B. f(x) has a minima in $({\alpha }_1,{\beta }_1)$ and a maxima in $(\alpha ,\beta )$
• C. f'(x) > 0 wherever defined
• D. f'(x) < 0 wherever defined

Answer 1

The correct option is A

f(x) has a maxima in $[{\alpha }_1,{\beta }_1]$ and a minima is $[\alpha ,\beta ]$

Clearly, f(x) is increasing in $({\alpha }_1,{\beta }_1)$
Clearly, f(x) has a maximum in $|{\alpha }_1,{\beta }_1|$ and a minima in $[\alpha ,\beta ]$, shown as