$\frac{\text{Quadratic}}{\text{Quadratic}}$ methods

Question

So, lets take up a question.
Show that $f(x)=\frac{x^2+34x-71}{x^2+2x-7}$ can never lie between 5 and 9.

MY ATTEMPT:
I assumed the function to be equal to k , then cross multiplied and got a single quadratic. Now solved for its discriminant to be real, got another quadratic in k , solved for it and got the answer.

Can someone explain(refer) shorter and better methods to solve such rational functions?

Any help is welcome.
Thanks.

Answer 1

The roots of $f'(x)=0$ are $x=1$ and $x=3$.

From second derivative test if $a$ is the critical point of $f(x)$ (i.e. $f'(a)=0$) then

• If $f''(a)>0$, $f$ has local minimum at $a$
• If $f''(a)<0$, $f$ has local maximum at $a$
• If $f''(a)=0$, test is inconclusive

we find $f''(1)=4>0$ and $f''(3)=-1<0$. So $f$ has local minimum at $(1,9)$ and local maximum at $(3,5)$.

Hence from here we can infer $f$ never lie between $5$ and $9$.