To determine the smallest degree of a polynomial that has five relative minima and six relative maxima, we first need to understand how the number of relative minima and maxima is related to the degree of the polynomial.
A polynomial of degree \( n \) can have at most \( n - 1 \) turning points (where the turning points include both relative minima and maxima). This means that the maximum number of turning points is determined by the degree of the polynomial.
In this case, we have:
- 5 relative minima
- 6 relative maxima
The total number of turning points is: \[ 5 + 6 = 11 \]
According to the relationship, a polynomial of degree \( n \) can have at most \( n - 1 \) turning points. Therefore, for our polynomial, we need: \[ n - 1 \geq 11 \] This leads us to: \[ n \geq 12 \]
Thus, the smallest degree of a polynomial that could have five relative minima and six relative maxima is \( n = 12 \).
So, the answer is 12.