The question asks about a fourth-degree polynomial function $f(x)$, and specifically, the maximum number of critical points it can have.

Key points to solve this:

1. Degree of the Polynomial: The polynomial is of degree 4, meaning its general form is: $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ where $a, b, c, d, e$ are constants.

2. Critical Points: Critical points occur where the first derivative $f'(x)$ equals zero.

3. First Derivative: The derivative of a 4th-degree polynomial is a 3rd-degree polynomial: $f'(x) = 4ax^3 + 3bx^2 + 2cx + d$ A third-degree polynomial can have at most 3 real roots (because the number of critical points corresponds to the number of real roots of $f'(x)$).

Therefore, the maximum number of critical points for a 4th-degree polynomial is 3.

The correct answer is: (c) 3.

Would you like more details or have any questions about the solution?

Here are 5 related questions for practice:

1. How do you find the critical points of a function in general?
2. What is the minimum number of critical points a 4th-degree polynomial can have?
3. How does the degree of the first derivative affect the number of critical points?
4. Can a polynomial of degree 3 have fewer than 3 critical points?
5. What is the significance of a critical point in determining the behavior of a function?

Tip: The critical points help identify where the function has local maxima, minima, or saddle points (inflection points).