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The problem requires identifying a function \( f(x) \) based on given characteristics and sketching the appropriate graph from the provided options.

### Information given:
- \( f(-5) = f(4) = 0 \): The function has roots at \( x = -5 \) and \( x = 4 \).
- \( f'(-0.5) = 0 \): The derivative is zero at \( x = -0.5 \), indicating a critical point (likely a local maximum or minimum).
- \( f'(x) < 0 \) for \( x < -0.5 \): The function is decreasing for \( x < -0.5 \).
- \( f'(x) > 0 \) for \( x > -0.5 \): The function is increasing for \( x > -0.5 \).

### Analysis:
1. Roots at \( x = -5 \) and \( x = 4 \) indicate the function crosses the x-axis at these points.
2. There is a local extremum at \( x = -0.5 \) because \( f'(-0.5) = 0 \). The function decreases to this point and increases afterward, indicating a local minimum at \( x = -0.5 \).
3. The function should be negative between the roots \( x = -5 \) and \( x = 4 \), except where it has local extrema.

### Conclusion:
The function will likely be a cubic or a quartic polynomial, and it must have these roots and behave accordingly. Looking at the graphs, the one that satisfies all these conditions is **the graph in the top left**.

### Reasoning for selecting the top left graph:
- The curve crosses the x-axis at \( x = -5 \) and \( x = 4 \), satisfying the root conditions.
- There is a local minimum at \( x = -0.5 \), as required by \( f'(-0.5) = 0 \), with the function decreasing to the left of \( x = -0.5 \) and increasing to the right of it.
- The graph shows the function decreasing for \( x < -0.5 \) and increasing for \( x > -0.5 \), aligning with the derivative information.

Would you like more details on how to sketch or analyze this type of function further? Here are some related questions you might find helpful:

1. How do we find the derivative of a function to confirm local extrema?
2. How do critical points help us determine the shape of a graph?
3. What is the role of concavity in identifying the type of extremum (minimum/maximum)?
4. How does knowing the roots of a function help in graphing it?
5. How can symmetry in a function's graph provide additional insights?

**Tip:** Always use the first and second derivatives of a function to analyze its behavior, including increasing/decreasing intervals and concavity.

Identify a function f that has the given characteristics. f(-5) = f(4) = 0; f'(-0.5) = 0, f'(x) < 0 for x < -0.5; f'(x) > 0 for x > -0.5. Sketch the function.

This problem requires identifying a function based on roots, critical points, and increasing/decreasing intervals. It uses concepts from calculus, including the first derivative test and polynomial function analysis. Suitable for Grades 11-12 or early calculus students.

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