Re: Shape of quartic functions



On Sep 8, 5:17 am, Albert <albert.xtheunkno...@xxxxxxxxx> wrote:
Quartics have a u shape unless they're of the form f(x) = x^4 + ax^2
which have a w shape. Correct?

A general quartic has a w-like shape (or m-like, if the leading
coefficient is negative). I say "like" because the two local minima
(maxima) may not be at the same value. There are also degenerate
cases, such as when the quartic is the (negative) square of a
quadratic that is always positive or negative. In this case it looks
like a parabola (although it isn't); this is not the only way to get a
u-shaped (n-shaped) graph.

What you should do is look at the cases of quartics whose derivatives
have 1, 2, or 3 distinct roots. If there is only 1 root, then it is u-
shaped. If there are 3, it is w-shaped. If there are 2 (meaning there
is a repeated root), then you get another shape, sort of like the top
of many cars. If there are 0, then you did something wrong, since a
cubic must have at least one root ;-)
.