Question

What is the smallest degree a polynomial with five relative minima and six relative maxima could have?
A. 30
B. 12
C. 11
D. 6

Answer 1

To find the smallest degree of a polynomial that can have a given number of relative minima and maxima, we can use the relationship between the number of local extrema (minima and maxima) and the degree of the polynomial.

Let \( m \) represent the number of relative minima and \( M \) represent the number of relative maxima. The total number of relative extrema is given by:

\[
E = m + M
\]

In this case, we have:

\[
m = 5 \quad \text{and} \quad M = 6
\]

Thus, the total number of relative extrema is:

\[
E = 5 + 6 = 11
\]

According to the properties of polynomials, a polynomial of degree \( n \) can have at most \( n - 1 \) relative extrema. Therefore, to accommodate \( 11 \) relative extrema, the degree \( n \) of the polynomial must satisfy:

\[
n - 1 \geq E
\]

Substituting for \( E \):

\[
n - 1 \geq 11
\]

Solving for \( n \):

\[
n \geq 12
\]

This means that the smallest degree \( n \) is \( 12 \).

Thus, the answer is:

\[
\boxed{12}
\]