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To solve this problem, we need to analyze the given function graph to determine the signs of the function values \(f(x)\), the first derivative \(f'(x)\), and the second derivative \(f''(x)\) at the specified points: \(x = -3\), \(x = 3\), \(x = 5\), and \(x = 9\).

### 1. \(f(x)\) - Function value:
   - \(f(x)\) indicates the height of the curve at the given \(x\)-value.
   
### 2. \(f'(x)\) - First derivative (slope of the function):
   - If the function is increasing, \(f'(x) > 0\) (positive slope).
   - If the function is decreasing, \(f'(x) < 0\) (negative slope).
   - If there is a horizontal tangent line (flat), \(f'(x) = 0\).

### 3. \(f''(x)\) - Second derivative (concavity of the function):
   - If the graph is concave up (like a cup), \(f''(x) > 0\).
   - If the graph is concave down (like a hill), \(f''(x) < 0\).
   - At an inflection point, \(f''(x) = 0\), where the concavity changes.

Let's analyze the graph step by step for each point:

### \(x = -3\):
- \(f(x)\): The function value is negative, as the curve is below the \(x\)-axis.
- \(f'(x)\): The slope is negative (the function is decreasing).
- \(f''(x)\): The graph is concave up (curving upwards), so \(f''(x) > 0\).

### \(x = 3\):
- \(f(x)\): The function value is negative (below the \(x\)-axis).
- \(f'(x)\): At \(x = 3\), the graph has a horizontal tangent, meaning \(f'(x) = 0\).
- \(f''(x)\): The graph is concave down at this point (curving downwards), so \(f''(x) < 0\).

### \(x = 5\):
- \(f(x)\): The function value is positive (above the \(x\)-axis).
- \(f'(x)\): The slope is negative (the function is decreasing).
- \(f''(x)\): The graph is concave up at this point (curving upwards), so \(f''(x) > 0\).

### \(x = 9\):
- \(f(x)\): The function value is positive (above the \(x\)-axis).
- \(f'(x)\): The slope is positive (the function is increasing).
- \(f''(x)\): The graph is concave up (curving upwards), so \(f''(x) > 0\).

### Summary Table:

| x  | \(f(x)\) | \(f'(x)\) | \(f''(x)\) |
|----|----------|-----------|------------|
| -3 |    -     |     -     |      +     |
|  3 |    -     |     0     |      -     |
|  5 |    +     |     -     |      +     |
|  9 |    +     |     +     |      +     |

Would you like further clarification on any part of this process?

Here are 5 related questions to deepen understanding:
1. How do inflection points relate to the second derivative \(f''(x)\)?
2. What are the key differences between concave up and concave down regions?
3. How does the first derivative help in identifying local maxima and minima?
4. Why is the second derivative zero at inflection points?
5. How can the graph of the function help us predict the behavior of higher-order derivatives?

**Tip**: Always check both the first and second derivatives when analyzing the shape and behavior of a function.

Fill in the table with +, -, or 0 for the nonlinear function shown in the graph. Determine the signs of f(x), f'(x), and f''(x) at x = -3, 3, 5, and 9.

Learn how to analyze the function values, the first and second derivatives from a nonlinear function graph. This problem helps understand increasing/decreasing behavior, concavity, and critical points for x = -3, 3, 5, and 9. Suitable for high school students in Grades 10-12.

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