Francisco's teacher asked him to describe the number of relative extrema a polynomial of degree five could have. He answered, "The polynomial will have exactly five relative extreme". Is Francisco correct or incorrect? Explain your answer.(1 point)

Question

Responses

Francisco is incorrect, because the number of extrema a polynomial could have is one less than the degree.
Francisco is incorrect, because the number of extrema a polynomial could have is one less than the degree.

Francisco is correct, because the degree is the exact amount of relative extrema a polynomial will have.
Francisco is correct, because the degree is the exact amount of relative extrema a polynomial will have.

Francisco is incorrect, because the polynomial could have more than five relative extrema.
Francisco is incorrect, because the polynomial could have more than five relative extrema.

Francisco is correct, because all polynomials have exactly five relative extrema.

Answer 1

Francisco is incorrect, because the number of extrema a polynomial could have is at most one less than the degree. A polynomial of degree five can have up to four relative extrema.

In general, a polynomial of degree $n$ can have at most $n-1$ relative extrema. This is because the first derivative of a polynomial (which is used to find critical points) is a polynomial of degree $n-1$, and thus can have at most $n-1$ real roots (i.e., critical points). Therefore, while a degree five polynomial could have as few as zero relative extrema, it can have a maximum of four relative extrema, not five.

Thus, the correct response is: "Francisco is incorrect, because the number of extrema a polynomial could have is one less than the degree."