# Re: Shape of quartic functions

*From*: The Qurqirish Dragon <qurqirishd@xxxxxxx>*Date*: Tue, 8 Sep 2009 06:27:35 -0700 (PDT)

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 ;-)

.

**References**:**Shape of quartic functions***From:*Albert

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