Above we have a balanced set of functions in the form

Y .= .(x⁴ + 8x³ + 19x² + 12x) times Q

Q varies from 1/8, the bluest curve -- to 2, the tallest curve.

Since the ratios between the coefficients stay constant, the functions stack their points of identity.

These being: the relative maximums and minimums, which stack on lines,

and the functions' natural roots, which stack in place.

asdf

The natural root is a term I use to describe the roots at the level the function crosses the Y axis.

Here the function crosses Y at 0, so there is no difference between the "natural root",

and the zero root -- or simply "the root" -- where the function equals 0.

If we added a numercial coefficient, (say 2): ..Y .= .x⁴ + 8x³ + 19x² + 12x + 2

It would cross Y at 2: and the natural roots would be those places it returned to 2 again.

Although generally speaking, we look for the zero root of an equation -- because we want to solve it --

mathematics is much more forthcoming in revealing the natural roots.

Here, roots are as they appear: 0, -1, -3, -4.

asdf

I speak of these functions as being balanced.

By that I mean they have left-right symmetry -- a rarity in mixed higher degree curves.

( I tailored them carefully for this occasion. )

I can't really say why they're balanced -- it would be a wonderful avenue to pursue...

The numerical difference between the coefficients is 7, 11, 7

The difference between the roots is 1, 2, 1

Math is saying something. .Once it's understood, it should be both coherant and unexpected.

asdf

Now let's take a look at the relative maxima and minima.

These being fourth degree curves, they are entitled to reverse themselves up to three times.

And do so splendidly.

There's a way to find the X values of these reversals, and it's quite simple -- but it comes with caveats.

First the method: . Solve the function you get by multiplying each coefficient by its exponent.

If there is a numerical coefficient -- leave it out, it will throw everything off.

asdf

Y .= .x⁴ + 8x³ + 19x² + 12x + 6

will have its maxima and minima at the 0 roots of

4x⁴ + 24x³ + 38x² + 12x

asdf

But!

1: There will be an extra solution, (a zero).

2: I do not trust the complex space answers this method returns --

The initial i values are red herrings, not to be considered.

The complex x values may be more subtly flawed.