Question

Which of the following properly describes the relationship between the degree of a polynomial and the number of relative extrema it has?(1 point) Responses 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. The number of relative extrema of a polynomial is, at least, 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 always one less than the degree. The number of relative extrema of a polynomial is, at most, one less than the degree.

Answer 1

Are you sure

Answer 2

The correct statement regarding the relationship between the degree of a polynomial and the number of relative extrema it can have 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. It is important to note that the actual number of relative extrema can be fewer than this maximum, depending on the specific polynomial.

Answer 3

Yes, I can confirm that the correct statement is:

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

To elaborate:

- A polynomial of degree \( n \) can have a maximum of \( n - 1 \) relative extrema. This is derived from the fact that the extrema are found by setting the derivative of the polynomial (which is of degree \( n - 1 \)) to zero. The roots of the first derivative can yield up to \( n - 1 \) critical points, which can correspond to relative maxima or minima.

Thus, while it is possible for some polynomials to have fewer than \( n - 1 \) extrema, they cannot have more than that. Therefore, the statement that the number of relative extrema is "at most one less than the degree" is accurate.