To solve this question, let's analyze the information given:

1. Degree of $f(x)$: Since $f$ is a polynomial function of degree 4, it has a maximum of 4 roots (or real solutions) by the Fundamental Theorem of Algebra.

2. Critical Points: The critical points of a function are found where its derivative $f'(x)$ is zero or undefined. Since $f(x)$ is a fourth-degree polynomial, its derivative $f'(x)$ will be a third-degree polynomial.

3. Number of Critical Points: A third-degree polynomial can have up to three real roots (solutions where $f'(x) = 0$), which correspond to the critical points of $f(x)$.

Therefore, the greatest number of critical points that $f(x)$ can have is 3.

Answer:

The correct answer is (c) 3.

Would you like a further explanation or have any questions?

Here are some related questions to explore:

1. How do we determine the degree of the derivative of a polynomial function?
2. What are critical points, and why do they matter in calculus?
3. How does the Fundamental Theorem of Algebra help in finding the maximum number of roots?
4. Can a fourth-degree polynomial have more than three turning points?
5. What is the relationship between the degree of a function and the number of its critical points?

Tip: For any polynomial function of degree $n$, the maximum number of critical points is $n-1$.