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Question
What is the smallest degree a polynomial with five relative minima and six relative maxima could have?(1 point)

Answer 1

To determine the smallest degree a polynomial can have given the number of relative minima and maxima, we can use the following reasoning:

For a polynomial of degree \( n \), the number of relative extrema (which includes both relative minima and relative maxima) is at most \( n - 1 \). This is because the extrema occur at critical points, which are found by taking the derivative of the polynomial and setting it to zero.

In this case, you have:

- Five relative minima
- Six relative maxima

This amounts to a total of:

\[
5 + 6 = 11 \text{ relative extrema}
\]

Since the maximum number of relative extrema for a polynomial of degree \( n \) is \( n - 1 \), we set up the inequality:

\[
n - 1 \geq 11
\]

Solving for \( n \):

\[
n \geq 12
\]

Thus, the smallest degree \( n \) such that the polynomial can have at least 11 relative extrema is:

\[
\boxed{12}
\]