How do you find the relative maximum of a given cubic formula using algebra?

Question

I know how to do it with caluculus. I just want to know how to do it with algebra as in no derivativation, integration, or limits...

The reason for me asking this is because an individual in a lower class recieved the same assignment as I did and he doesn't have the calculus knowledge. If he used Calculus it would seem as if he either copied from a upper level classmate which can be considered cheating or be advided to move up a level if he does indeed have a prior knowledge of Calculus. I'm currently trying to derive the formula myself, I'm just asking since I'm rather busy at the moment.

Oh and by the way, this individual doesn't have a calculator so the graphing method isn't vaid considering the value is rational only in the sense that it can be expressed in terms of a/b. The answer I would suppose could only be obtained algebraicly.

Answer 1

make a table of values, this is the most precise way when you don't have a graphing calculator and you suck at graphing.

Answer 2

I don't entirely understand this question. I understand it, but I don't know why it would be asked, given that this seems to be a fairly reasonable and straightforward answer to someone that knows calculus.

1. Get someone to differentiate the equation
Ax^3 + Bx^2 + Cx + D
with respect to x (call this formula 1). differentiate again (call this formula 2)
2. Use quadradic formula on formula 1 by setting it to zero.
3. use formula 2 to see which (if any) answer is a relative max by conducting sign test

This requites no calculus except for Step 1. Once you've got the equations and apply the formulas you could make it so easy a 6th grader should be able to do it. (that is, assuming they could apply formulas).

If you're asking how one could derive something without calculus, I must admit I'm rather stumped, although I wouldn't be suprised if the knowledge of how to get the relative maxima of a cubic was derived before there was any understanding of calculus.

Answer 3

you can plot it on a graph and visually identify the inflection points

if things like that could be done in a straightforward way without calculus, it wouldn't have been necessary to invent calculus

Answer 4

yup i agree.. a graph is one of the best choices available if derivatives are not used.

Answer 5

Graph it.