Question

f(x) = x^5 - 3x^3 -x^2 -4x - 1

How do you graph this? I found the first derivative and it is not factorable so I can't really plug it in the original equation and find the minimum and stuff like that. Do I just use the 2nd derivative instead?

Answer 1

First thing to notice is that with all of the coefficients being the same order of magnitude, everywhere except very close to x = 0 this is going to look essentially like the graph of \(y = x^5\). Take x = 10, for example: polynomial evaluates to 96859, first term evaluates to 100000. The first term is (1/3)x^2 times the magnitude of the next largest term, so it really swamps everything.

Answer 2

While it may not be convenient to solve the first derivative = 0 by hand, you can certainly do it with electronic help. I would just compute a handful of points to get some idea of the overall behavior, maybe x = +- 100, 10, 5, 3, 2, 1, 0 and having gotten some idea of the overall shape, find some points in between where the 1st and 2nd derivatives suggest there might be interesting action. I'm all for using computers to do this sort of thing once you know how to do it yourself and have enough experience to recognize that the fancy graph you've just gotten doesn't look anything like what it should because you've entered the equation wrong or made some other blunder.