Given the function $f(x) = x^6 - x^2 + 9$, we can determine the following:

1. Maximum Number of Real Zeros:

The maximum number of real zeros corresponds to the degree of the polynomial, which is the highest power of $x$. Since the degree of $f(x)$ is 6, the function can have up to 6 real zeros. However, we must note that not all of these zeros must be real; some could be complex.

2. Maximum Number of $x$-Intercepts:

The maximum number of $x$-intercepts corresponds to the number of real zeros of the function. As mentioned, the function can have up to 6 $x$-intercepts, assuming all zeros are real.

3. Maximum Number of Turning Points:

The maximum number of turning points (local maxima and minima) is given by $n - 1$, where $n$ is the degree of the polynomial. For $f(x) = x^6 - x^2 + 9$, the degree is 6, so the function can have at most $6 - 1 = 5$ turning points.

Summary:

1. Maximum number of real zeros: 6
2. Maximum number of $x$-intercepts: 6
3. Maximum number of turning points: 5

Would you like more details or further clarifications on this?

5 Related Questions:

1. How do we determine if a polynomial function has complex roots?
2. Can a polynomial with real coefficients have an odd number of turning points?
3. What happens when all the zeros of a polynomial are complex?
4. How do the multiplicities of roots affect the graph of a polynomial?
5. What is the role of the leading coefficient in determining the end behavior of a polynomial?

Tip: The number of turning points is always one less than the degree of the polynomial, but it doesn't guarantee that all turning points are real.