Question Given that the polynomial $$ f(x) $$ has degree 9 , which of the following most accurately describes the number of turning points of $$ f(x) $$ ? Select the correct answer below. The graph of $$ f(x) $$ has at most 10 turning points. The graph of $$ f(x) $$ has at most 9 turning points. The graph of $$ f(x) $$ has at most 8 turning points. The graph of $$ f(x) $$ has at least 8 turning points. The graph of $$ f(x) $$ has at least 10 turning points. The graph of $$ f(x) $$ has at least 9 turning points.

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To determine the number of turning points of a polynomial function $$ f(x) $$ of degree $$ n $$, we can use the following rule:

- A polynomial of degree $$ n $$ can have at most $$ n - 1 $$ turning points.

Given that the polynomial $$ f(x) $$ has a degree of 9, we can calculate the maximum number of turning points:

$$
\text{Maximum turning points} = n - 1 = 9 - 1 = 8
$$

Now, let's analyze the options provided:

1. The graph of $$ f(x) $$ has at most 10 turning points. (Incorrect, as it can have at most 8 turning points)
2. The graph of $$ f(x) $$ has at most 9 turning points. (Incorrect, as it can have at most 8 turning points)
3. The graph of $$ f(x) $$ has at most 8 turning points. (Correct)
4. The graph of $$ f(x) $$ has at least 8 turning points. (Incorrect, as it can have fewer than 8 turning points)
5. The graph of $$ f(x) $$ has at least 10 turning points. (Incorrect)
6. The graph of $$ f(x) $$ has at least 9 turning points. (Incorrect)

Thus, the most accurate description of the number of turning points of $$ f(x) $$ is:

**The graph of $$ f(x) $$ has at most 8 turning points.**
The graph of $$ f(x) $$ has at most 8 turning points.

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