Question

the graph of a polynomial function of degree n has, at most, ______ turning points. the graph of a polynomial function of degree n has, at most, turning points.

Explanation:

Step1: Recall the relationship between degree and turning points

For a polynomial function \(y = f(x)\) of degree \(n\), the derivative \(y'=f'(x)\) is a polynomial of degree \(n - 1\).

Step2: Use the fact about the roots of the derivative

The turning - points of \(y = f(x)\) occur where \(y'=f'(x)=0\). A polynomial of degree \(n-1\) has at most \(n - 1\) real roots. Each real root of \(y'=f'(x)\) (where the sign of \(y'\) changes) corresponds to a turning - point of \(y = f(x)\).

Answer:

\(n - 1\)