Bezier Cubic Curves for six points

Question

Given six points in $\Bbb R^2$: $(-1,0),(-1,1),(-1,2),(1,0),(1,1),(1,2)$.

How to prove that there is not Cubic Bezier curve that cross all these points?

The picture of the points

Answer 1

I think it is possible, if we're allowed to interpolate the six points in any order we choose:

Answer 2

It is possible for a cubic polynomial to take the values $-1$ and $1$ three times, when it has two extrema. But only the sequence $-1,1,1,-1,-1,1$ (increasing, decreasing, increasing) or its opposite are possible.

Note that on $y$, the sequence $0,1,2,0,1,2$ is possible.