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Let's break down the problem using the given graph of \( f''(x) \), which represents the second derivative of \( f(x) \), to determine the concavity of \( f \) and locate its inflection points.

### Part (a) - Concavity:
- \( f(x) \) is concave **upward** where \( f''(x) > 0 \) (i.e., when the graph of \( f''(x) \) is above the x-axis).
- \( f(x) \) is concave **downward** where \( f''(x) < 0 \) (i.e., when the graph of \( f''(x) \) is below the x-axis).

From the graph:
- \( f''(x) \) is **negative** on the interval \( (-\infty, 2) \), so \( f \) is concave **downward** on \( (-\infty, 2) \).
- \( f''(x) \) is **positive** on the interval \( (2, \infty) \), so \( f \) is concave **upward** on \( (2, \infty) \).

### Part (b) - Inflection Points:
- The inflection points occur where \( f''(x) = 0 \) and changes sign. From the graph, \( f''(x) = 0 \) at \( x = 2 \) (where the curve crosses the x-axis). This is the inflection point because \( f''(x) \) changes from negative to positive at this point.

Thus, the x-coordinate of the inflection point is \( x = 2 \).

### Summary:
- **Concave downward** on \( (-\infty, 2) \).
- **Concave upward** on \( (2, \infty) \).
- The **inflection point** occurs at \( x = 2 \).

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

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Here are 5 related questions:
1. How do you find the concavity of a function if only \( f'(x) \) is given?
2. What is the significance of the second derivative in determining concavity?
3. How can you verify an inflection point using the first derivative \( f'(x) \)?
4. What happens to a function at a point where \( f''(x) = 0 \) but the concavity does not change?
5. How would the concavity of \( f(x) \) affect the shape of the graph of \( f(x) \)?

**Tip:** The sign of the second derivative gives key insights into whether a function is concave upward or downward, but always check for a change in concavity when identifying inflection points.

On what intervals is f concave upward or concave downward? What are the x-coordinates of the inflection points of f?

This calculus problem focuses on identifying the intervals where a function is concave upward or downward by analyzing its second derivative. Learn how to locate inflection points where the concavity changes, using graphical information. Suitable for students in College Calculus.

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