Question

lf $x$ is real, then the greatest and least values of the expression $\frac{x^2-2x+2}{x^2+3x+9}$ are respectively

• A. $2$ and $-\frac{2}{27}$
• B. $2$ and $\frac{2}{27}$
• C. $2$ and $\frac{1}{27}$
• D. $2$ and $3$

Answer 1

The correct option is B $2$ and $\frac{2}{27}$

$y=\frac{x^2-2x+2}{x^2+3x+9}$

$y{x}^2+3xy+9y=x^2-2x+2$

$x^2(y-1)+x(3y+2)+9y-2=0$

$D>0$

$(3y+2{)}^2-4(9y-2\left)\right(y-1)>0$

$9y^2+4+12y-4(9y^2-9y-2y+2)>0$

$36y^2-44y+8-9y^2-4-12y<0$

$27y^2-56y+4<0$

$(y-2)(y-\frac{2}{27})<0$

$\frac{2}{27}<y<2$