Max/Min values of quadratic functions of the form of $p/q$

Question

I want to know, how can we compute the max and min values of quadratic functions in the $p/q$ form. By $p/q$ , I mean, $$\frac{ax^2+bx+c}{kx^2+mx+r}$$ I know, that we can let the whole expression as $y$ and then create bounds on the discriminant, which would give us the answer. I want to know some other ways to do so, I tried calculus but it ended up becoming messy. I believe I can use polar form, but I am not sure how to do so. Any help would be appreciated.

Answer 1

There is no simple answer as it depends entirely in the values of $a,b,c,k,m,r$.

If you are (and I assume you are) working in $\mathbb R$, then your best bet is this:

First, look if you can simplify the expression. Look for the roots of the denominator, and if the numerator shares a root $x_0$, then both the denominator and numerator are multiples of $(x-x_0)$, and you can rewrite the expression as $\frac{Ax+B}{Cx+D}$.

Second, assuming there are no shared roots, look at the denominator's roots.

1. If there are two roots, then the entire expression is unbounded because it has a bidirectional pole at either root.
2. If there is a single root, then the expression is unbounded either below or above depending on the sign of the numerator and denominator near the root (it's a single-directional pole). Calculating the other bound will only be possible using derivatives.
3. If there is no root, then you are out of luck and derivatives are the only way.