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The problem involves using the **first derivative test** to determine maximum and minimum points of a function \( f(x) \). Let me guide you through it.

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### **First Derivative Test**
The first derivative test uses the sign of \( f'(x) \) (slope of \( f(x) \)) to determine where \( f(x) \) has **local minima** or **maxima**.

1. **If \( f'(x) \) changes sign:**
   - From positive to negative → Local maximum at that point.
   - From negative to positive → Local minimum at that point.

2. **If \( f'(x) \) does not change sign** → No extremum exists at that point.

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### Given Table Setup
The table is organized to analyze the behavior of \( f'(x) \) at points \( x < c \), \( x = c \), and \( x > c \), where \( c \) is a critical point.

- **Row 1:** \( f'(x) \) (First derivative) behavior.
- **Row 2:** Slope interpretation based on \( f'(x) \).

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### Completing the Table
1. At \( x < c \): Analyze whether \( f'(x) \) is **positive (increasing)** or **negative (decreasing)**.
2. At \( x = c \): \( f'(x) = 0 \) (Critical point).
3. At \( x > c \): Check if \( f'(x) \) is positive or negative again.

#### Example Filling:
| \( x \)        | \( x < c \)       | \( x = c \) | \( x > c \)       |
|----------------|-------------------|------------|-------------------|
| \( f'(x) \)    | \( + \) or \( - \) | \( 0 \)    | \( - \) or \( + \) |
| Slope          | Increasing/Decreasing | Flat      | Decreasing/Increasing |

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### To Determine:
- **Maximum point:** \( f'(x) \) changes **+ to -** at \( x = c \).
- **Minimum point:** \( f'(x) \) changes **- to +** at \( x = c \).

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Let me know if you need further clarification on completing the table or solving a specific function. Would you like me to provide an example with a function?  

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### Related Questions:
1. What is the significance of critical points in the first derivative test?
2. How can we identify an inflection point using derivatives?
3. How does the second derivative test differ from the first derivative test?
4. What happens if \( f'(x) \) does not exist at a point \( x = c \)?
5. How do you sketch \( f(x) \) based on \( f'(x) \)?

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### Tip:
Always check the sign of \( f'(x) \) **around critical points** to ensure you correctly classify maxima or minima.

Determine (if exist) minimum, maximum points using first derivative test.

Learn how to determine the maximum and minimum points of a function using the first derivative test. This step-by-step guide explores critical points, sign changes of the derivative, and slope analysis. Perfect for students studying calculus in Grades 11-12.

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