Probability that a cubic has two turning points

Most cubic polynomials with real coefficients have two turning points, a local maximum and a local minimum.

But how do you quantify “most”? Here’s how one author did it [1].

Start with the cubic polynomial x³ + ax² + bx + c Since multiplying a polynomial by a nonzero constant doesn’t change how many turning points it has, we might as well assume the leading coefficient is 1.

In his paper, Robert Fakler assumes a, b, and c are chosen randomly from an interval [-k, k].

He shows that for k ≤ 3, the probability that the polynomial has two turning points is p = (9 + k)/18.

For k ≥ 3, the probability is p = 1 – √(3/k) / 3 and so as k → ∞,  p → 1.

[1] Robert Fakler.

Do Most Cubic Graphs Have Two Turning Points? The College Mathematics Journal, Vol.

30, No.

5 (Nov.

, 1999), pp.


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