5

Some facts about cubic functions and quartic functions motivate this question:

  1. Every cubic function $f$ has exactly one inflection point $P$, and the graph $y=f(x)$ is symmetric about $P$. In particular, if $f$ has two local extrema $A$ and $B$, then $P$ is the midpoint of $AB$. Phrased in another way, the ratio $\frac{PA}{PB}$ is an invariant.

  2. A quartic function has either $0$ or $2$ inflection points. Suppose it has two inflection points $P$ and $Q$. The secant line passing through these two points will intersect the curve at two other points $A$ and $B$. Assume these four points are in the order $A, P, Q, B$ from left to right, then $AP:PQ:QB = 1:\frac{1+\sqrt{5}}{2}:1$. Phrased in another way, the ratios $\frac{AP}{PQ}$ and $\frac{AP}{QB}$ are invariants. There is also a fact about the areas bounded between the secant line and the curve.

Now consider a quartic function with three local extrema (and hence two inflection points). I wonder if there exists any invariant concerning the distances between these $5$ points, or their coordinates, or perhaps some areas bounded between lines and the curve. I'd be glad to see any such result.

Joseph
  • 580

1 Answers1

3

I've done some computations and found some relations among the slopes, so I decided to post this as an answer, which can hopefully attract others to explore the relations among the lengths and areas as well.

Suppose $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ with $a \neq 0$ has three local extrema $A, O, B$ and two inflection points $P, Q$. Label these five points $A, P, O, Q, B$ from left to right.

First, let me point out the following trivial facts by virtue of Vieta's formulas: the sum of all the (real and imaginary) roots of $f$ is $-\frac{b}{a}$, the sum of the $x$-coordinates of $A, O$ and $B$ is $-\frac{3b}{4a}$, and the sum of the $x$-coordinates of $P$ and $Q$ is $-\frac{b}{2a}$, so in particular \begin{equation} \tag{1} \frac{x_A+x_O+x_B}{x_P+x_Q} = \frac{3}{2}. \end{equation} In other words, the average of the $x$-coordinates of the critical points equals the average of the $x$-coordinates of the inflection points.

Since here we are concerned only with slopes (and similarly for lengths and areas), WLOG we can assume $O$ is the origin by performing a translation. A necessary (but not sufficient) condition for this is $f(0) = 0$ and $f'(0) = 0$, so that $d = e = 0$.

Then $$f(x) = ax^4 + bx^3 + cx^2 = x^2(ax^2+bx+c),$$ $$f'(x) = 4ax^3 + 3bx^2 + 2cx = x(4ax^2+3bx+2c),$$ $$f''(x) = 12ax^2 + 6bx + 2c = 2(6ax^2 + 3bx + c).$$

The critical points are $$O(0, 0),$$ $$A\Bigg(\frac{-3b-\sqrt{9b^2-32ac}}{8a}, \frac{\Big(3b+\sqrt{9b^2-32ac}\Big)^2 \Bigl(-3b^2+16ac-b\sqrt{9b^2-32ac}\Bigl)}{2048a^3}\Bigg),$$ $$B\Bigg(\frac{-3b+\sqrt{9b^2-32ac}}{8a}, \frac{\Big(3b-\sqrt{9b^2-32ac}\Big)^2 \Bigl(-3b^2+16ac+b\sqrt{9b^2-32ac}\Bigl)}{2048a^3}\Bigg),$$

and the inflection points are $$P\Bigg(\frac{-3b-\sqrt{9b^2-24ac}}{12a}, \frac{\Big(3b+\sqrt{9b^2-24ac}\Big)^2 \Bigl(-3b^2+20ac-b\sqrt{9b^2-24ac}\Bigl)}{3456a^3} \Bigg),$$ $$Q\Bigg(\frac{-3b+\sqrt{9b^2-24ac}}{12a}, \frac{\Big(3b-\sqrt{9b^2-24ac}\Big)^2 \Bigl(-3b^2+20ac+b\sqrt{9b^2-24ac}\Bigl)}{3456a^3} \Bigg).$$

Denote by $m_{XY}$ the slope of the line passing through $X$ and $Y$, and by $m_X$ the slope of the tangent line at $X$. Then

$$m_{AB} = \frac{9b^3-32abc}{64a^2},$$ $$m_{PQ} = \frac{b^3-4abc}{8a^2},$$

which implies that $m_{AB} - m_{PQ} = \frac{b^3}{64a^2}$.

$$m_{OP} + m_{OQ} = \frac{3b^3-14abc}{24a^2},$$ $$m_{OA} + m_{OB} = \frac{9b^3-40abc}{64a^2},$$ $$m_P + m_Q = \frac{b^3-4abc}{4a^2}.$$

Each of the above $5$ quantities is a linear combination of $\frac{b^3}{a^2}$ and $\frac{bc}{a}$, so it is easy to find many ratios that are independent of the coefficients of $f$. The difficulty lies in finding simple and elegant ones such as the following two:

\begin{equation} \tag{2} m_{PQ} = \frac{1}{2}(m_P + m_Q). \end{equation} \begin{equation} \tag{3} \frac{m_{AB}-m_{OA}-m_{OB}}{m_{PQ}-m_{OP}-m_{OQ}} = \frac{3}{2}. \end{equation}

I'm looking forward to more insights from this community regarding other invariant quantities.

Joseph
  • 580