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:
\[ \boxed{12} \]