Yo lovely Mathpeople!
Intro
For the past several months, I've been playing around with a particular math problem, but since I've been unable to make much progress on it (and my math-versed roommate hasn't been heard of in weeks) and since it's simply too frustrating to leave completely, I decided to set aside my pride and ask for help.
Perhaps worst of all is the fact that I've tried 'dumbing down' the function and using fixed numbers instead of variables, -as that gives a basic type of 'find function f' question-, but even then it seems like it's been too long since high school.
Also, I'm the type of person that likes to try and apply multiple possibilities and deduce from answers, which is not really helpful when working with these kinds of problems.... sigh.
To the math!
Problem description
Basically, I want to find the funtion f(x); ultimately in terms of k, l (and x).
Given are the tops: -2,k/l; -1, k-l; (0,0 (point, not top)); 1, k+l and 2,k*l.
Ideally, the function runs from -infinity to infinity.
My own efforts
Since f(-larger)=-larger and f(larger)=larger (xD sorry for raping math convention by formulating it like this), f(0)=0 and four tops, the general form of the formula will be:
f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex.
Hell, if those are tops, the derivative at those points must be 0.
So f'(x) must equal 0 at -2, -1, 1 and 2.
Dumbing it down
Here's when I started to solve it with numbers, but somehow I've even forgotten how to do this :/..
Let's say k = 7 and l = 4.
That would make the tops: -2, 7/4; -1, 3; 1, 11 and 2, 28.
The derivative of fdumb(x) would have zeroes at those x's as well.
Then fdumb'(x) equals some form of (x-2)(x-1)(x+1)(x+2).
(And here's where I died. Brain-freeze or whatever xD)
So, all help is appreciated. Complete solution(s) to the problem are as welcome as a small nudge in the right direction of how to continue with the dumbed version.
Thanks in advance!