What is the maximum number of critical points a third degree polynomial can have

If it is more than 2 please provide an example.

third degree polynomial can have up to 2 critical points, unless something like this can be considered 3rd degree polynomial (2x^3+x^2+5)/(x-3) in this case there can be 3 critical points , 2 for where the derivative is zero and one for when the derivative is undefined. Hope I helped more than I confused you.

Two

The two points where f '(x) crosses the x-axis are called critical values for the function f(x)
The tho the corresponding points on the graph of f(x) are called critical points of the curve.

2, because the derivative of a cubic is a quadratic, which has, at most, 2 zeros.