For which real α does the curve y = x4 + 9x3 + α x2 + 9x + 4 contain four collinear points?
Answer: α < 30.375.
We look at the second derivative y'' = 12x2 + 54x + 2α = 12(x + 2.25)2 + 2(α - 30.375). This has two zeros iff α < 30.375.
If α > 30.375, then y'' has no zeros and hence y' is strictly monotonically increasing. Hence no line cuts the curve in more than two points (if a line gradient k cut the curve at x = a, b and c, with a < b < c, then there would be a point between a and b with gradient k and another between b and c, contradicting the strict monotonicity of y'.].
If α < 30.375, the y'' has two zeros either side of x = 2.25. Hence the curve has two points of inflection either side of x = 2.25. The line through these two inflection points will cut the curve in four points. [Suppose the two points are at x = a and x = b with a < b. The line has a smaller gradient than the tangent at the point of inflection at x = a and hence cuts the curve again for x < a. Similarly it has larger gradient than the tangent at the point of inflection at x = b and hence cuts the curve again for x > b.]
It remains to consider the case α = 30.375. In this case y'' has a single zero at x = 2.25. In this case there is no point of inflection. The gradient is monotonic and hence a line can only cut the curve in two points. [We cannot have degeneracy with infinitely many solutions because we are dealing with a polynomial which has at most 4 solutions.]
© John Scholes
12 Dec 1998