Question

Consider a rational function $f(x)=\frac{x^2-3x-4}{x^2-3x+4}$ and a quadratic function $g(x)=x^2-(b+1)x+b-1$, where $b$ is a parameter.
If both roots of the equation $g(x)=0$ are greater than $-1$, then b lies in the interval.

• A. $(-\infty ,-2)$
• B. $\left(-\infty ,\frac{-1}{4}\right)$
• C. $(-2,\infty )$
• D. $\left(\frac{-1}{2},\infty \right)$

Answer 1

Answer: (D)

Different possibilities are as follows :

If both roots of $g(x)=0$ are greater than $-1$, then 3 conditions should be satisfied simultaneously.
(1) $D\ge 0$
(2) $\frac{-B}{2A}>-1$
(3) $g(-1)>0$
Now, $(1)\Rightarrow (b+1{)}^2-4(b-1)\ge 0$
$\Rightarrow b^2-2b+5\ge 0\forall b\in R$
$(2)\Rightarrow \frac{b+1}{2}>-1\Rightarrow b>-3$
and $(3)\Rightarrow 1+(b+1)+(b-1)>0$
$\Rightarrow b>-\frac{1}{2}$
hence from $(1)\cap (2)\cap (3)$, we get
$b\in \left(\frac{-1}{2},\infty \right)$