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Answer all of the questions below about the function f(x) graphed below when x=2.

To analyze the graph of the function \( f(x) \) at \( x = 2 \), let's extract the relevant information.

### 1. **Value of \( f(2) \):**
   - Look at the graph where \( x = 2 \). The corresponding \( y \)-value is the value of \( f(2) \).
   - **\( f(2) = 0 \)** (since the graph intersects the x-axis at \( x = 2 \)).

### 2. **Slope of the Tangent at \( x = 2 \) (Derivative \( f'(2) \)):**
   - The slope of the tangent line at a point represents the derivative at that point.
   - At \( x = 2 \), the graph appears to be flat (the curve is neither increasing nor decreasing).
   - **\( f'(2) = 0 \)** (the tangent is horizontal at \( x = 2 \)).

### 3. **Concavity at \( x = 2 \):**
   - The concavity of the graph can be inferred from whether the graph is curving upwards (concave up) or downwards (concave down).
   - Around \( x = 2 \), the graph curves upwards.
   - **Concave Up** at \( x = 2 \), so \( f''(2) > 0 \).

### 4. **Local Maximum or Minimum at \( x = 2 \):**
   - A horizontal tangent where the concavity changes indicates a local extremum.
   - Since the graph is concave up at \( x = 2 \), and the derivative changes from negative to positive, **\( x = 2 \) is a local minimum**.

### 5. **Is \( x = 2 \) an Inflection Point?**
   - An inflection point occurs where the concavity changes.
   - The concavity is positive on both sides of \( x = 2 \), so there is no inflection point at \( x = 2 \).

### Summary of Findings at \( x = 2 \):
- \( f(2) = 0 \)
- \( f'(2) = 0 \)
- Concave Up ( \( f''(2) > 0 \) )
- Local Minimum at \( x = 2 \)
- No Inflection Point

Would you like more details or have any questions?

### Relative Questions:
1. What is the derivative of \( f(x) \) at other critical points such as \( x = 4 \) and \( x = -2 \)?
2. How does the concavity change around the point \( x = -4 \)?
3. What is the significance of inflection points in the graph of \( f(x) \)?
4. How would you determine the intervals where \( f(x) \) is increasing or decreasing?
5. What are the x-intercepts and y-intercepts of \( f(x) \)?

### Tip:
When analyzing graphs, always consider the derivative and concavity to determine the nature of critical points and inflection points.

Explore the analysis of function f(x) at x = 2, covering its value, slope (derivative), concavity (second derivative), and local extremum. Understand how to determine these critical points using calculus concepts and theorems like the Mean Value Theorem and Second Derivative Test.

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