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:
\[ \boxed{12} \]
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