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

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
$\underset{\left|x\right|\to \infty }{\lim }f\left(x\right)=\frac{a{x}^2+bx+c}{d{x}^2+ex+f}=f\left(x\right)=\frac{a}{d}$
$\Rightarrow bdx+cd=aex+af\Rightarrow (ae-bd)x=cd-af$
So, ae $\ne$  bd