To determine the smallest degree a polynomial can have given that it has five relative minima and six relative maxima, we can use the relationship between the number of relative extrema and the degree of the polynomial.
-
For a polynomial of degree \( n \), the maximum number of relative minima and maxima (extrema) it can have is \( n-1 \). This is because the derivative of a polynomial of degree \( n \) will be a polynomial of degree \( n-1 \), which can have \( n-1 \) roots (indicating potential relative extrema).
-
In this case, we have 5 relative minima and 6 relative maxima. Therefore, the total number of relative extrema is: \[ 5 ,(\text{minima}) + 6 ,(\text{maxima}) = 11 ,(\text{extrema}). \]
-
Based on our earlier observation, to accommodate 11 extrema, the degree \( n \) must satisfy: \[ n - 1 \geq 11. \] Solving for \( n \), we find: \[ n \geq 12. \]
Thus, the smallest degree a polynomial with five relative minima and six relative maxima could have is 12.
The answer is 12.
1 answer