Question

If $[a, b]$ is the range of the function $\frac{x+2}{2 x^{2}+3 x+6}$ for $x \in R$, then

• A. $a < 0, b < 0$
• B. $a < 0, b > 0$
• C. $a > 0, b > 0$
• D. $a > 0, b < 0$

Answer 1

Answer: (B)

$[a, b]$ is range of $\frac{x+2}{2 x^{2}+3 x+6}$ and $x \in R$
Let $y=\frac{x+2}{2 x^{2}+3 x+6}$
$\Rightarrow 2 y x^{2}+3 x y+6 y=x+2$
$\Rightarrow 2 y x^{2}+(3 y-1) x+6 y-2=0$
$x \in R \text { So, } D \geq 0$
$\Rightarrow (3 y-1)^{2}-4(6 y-2)(2 y) \geq 0$
$\Rightarrow -39 y^{2}+10 y+1 \geq 0$
$\Rightarrow 39 y^{2}-10 y-1 \leq 0$
$\Rightarrow (3 y-1)(13 y+1) \leq 0$
$\Rightarrow y \in\left[-\frac{1}{13}, \frac{1}{3}\right]$
So, $a=-\frac{1}{13}, b=\frac{1}{3}$
$\therefore a < 0, b > 0$