Let  $f\left(x\right)=\frac{(a{x}^2+bx+c)}{(d{x}^2+ex+f)}$. If f(x)is not a constant function and both minimum and maximum values of  $f\left(x\right)$ exist then

For  $f\left(x\right)$ to be bounded, its denominator should never become zero. So, its discriminant must be negative. Also, for some finite  $x,f\left(x\right)$ must be equal to 

${\lim }_{\left|x\right|\to \infty }f\left(x\right) i.e, \frac{a{x}^2+bx+c}{d{x}^2+ex+f}=\frac{a}{d}$

$\Rightarrow bdx+cd=aex+af\Rightarrow (ae-bd)x=cd-af$

$So, ae\ne bd$