To determine the maximum number of turning points for the function $f(x) = 2x^3 + 3x^2 + 3$, we can use the following rule:

For a polynomial function of degree $n$, the maximum number of turning points is $n - 1$.

Solution:

1. The degree of $f(x) = 2x^3 + 3x^2 + 3$ is 3 (since the highest power of $x$ is 3).
2. Applying the rule, the maximum number of turning points is $3 - 1 = 2$.

Answer:

The graph of $f$ has a maximum of 2 turning points.

Would you like further details on the solution or have any other questions?

Related Questions:

1. What is the degree of a polynomial and how does it affect the shape of the graph?
2. How do you find the actual turning points of a polynomial function?
3. What is the difference between a turning point and an inflection point?
4. How does the leading coefficient influence the end behavior of a polynomial graph?
5. How can we determine whether each turning point is a local maximum or minimum?

Tip:

To locate actual turning points, take the derivative of the function and find where it equals zero. Then, use the second derivative test to classify each turning point as a maximum or minimum.