What is the maximum number of relative extrema that a sixth degree polynomial degree polynomial can have?
The maximum number of relative extrema that a sixth degree polynomial can have is 5.
To find the maximum number of relative extrema that a sixth-degree polynomial can have, we need to consider the behavior of its derivative.
The number of relative extrema is equal to the number of changes in concavity of the original function. This happens when the derivative of the function changes sign.
A sixth-degree polynomial has a fifth-degree derivative. The maximum number of times the sign of the derivative can change is equal to the maximum number of roots that the fifth-degree polynomial can have. Since a fifth-degree polynomial can have at most five roots, the sixth-degree polynomial can have a maximum of five changes in concavity, and hence a maximum of five relative extrema.