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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.

mathsisfun
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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.
5xum
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  • Thanks for your explanation, yes the question indeed is vague, but what if all $a,b,c,k,m,r$ are integers? Will that help us get a exact solution? – mathsisfun Oct 05 '23 at 15:52
  • @mathsisfun No, even if they are all integers, you can have a situation where you have to look at derivatives. For exmple, $\frac{x^2+x+1}{x^2+1}$ is such a function. Its maximum is at $x=1$, and really, derivatives are the quickest and simplest way to calculate that. – 5xum Oct 06 '23 at 03:18