Question

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