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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**.

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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).

If the function f is a polynomial function of fourth degree and its domain is R, then the greatest number of the critical points of the function f(X) is:

Explore how to determine the maximum number of critical points of a fourth-degree polynomial. This question involves polynomial functions, derivatives, and critical points, suitable for students in Grades 11-12 or early college.

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