Which of the following properly describes the relationship between the degree of a polynomial and the number of relative extrema it has?(1 point)

Question

Responses

The number of relative extrema of a polynomial is always one less than the degree.
The number of relative extrema of a polynomial is always one less than the degree.

The number of relative extrema of a polynomial is, at most, one less than the degree.
The number of relative extrema of a polynomial is, at most, one less than the degree.

The number of relative extrema of a polynomial is equal to the degree.
The number of relative extrema of a polynomial is equal to the degree.

The number of relative extrema of a polynomial is, at least, one less than the degree.

Answer 1

The correct description of the relationship between the degree of a polynomial and the number of relative extrema it has is:

The number of relative extrema of a polynomial is, at most, one less than the degree.

This means that if a polynomial has a degree \( n \), it can have up to \( n - 1 \) relative extrema. This is based on the fact that the relative extrema are found by solving the derivative of the polynomial, which is a polynomial of degree \( n-1 \). Therefore, it can have at most \( n-1 \) critical points, which correspond to potential relative extrema.