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 .
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.
 Robert Fakler.
Do Most Cubic Graphs Have Two Turning Points? The College Mathematics Journal, Vol.
, 1999), pp.