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 the equations $x^2$ + bx + c = 0 and $x^2+b_{1}x+c_1=0$ do not have real roots, then

• A. f'(x) = 0 has real and distinct roots
• B. f'(x) = 0 has real and equal roots
• C. f'(x) = 0 has imaginary roots
• D. nothing can be said

Answer 1

The correct option is A

f'(x) = 0 has real and distinct roots

f(x) has one of the two graphs
$\Rightarrow$ f'(x) = 0 has real and distinct roots.