https://file.aiquickdraw.com/mathbot/file/17270702707413xlwwxp3.jpg

The problem asks to sketch the graph of a function \( f(x) \) based on the following conditions:

1. \( f(0) = 0 \)
2. \( f'(0) = 3 \)
3. \( f'(1) = 0 \)
4. \( f'(2) = -1 \)

These conditions give important information about the function and its derivatives at specific points:

- **\( f(0) = 0 \)**: The graph passes through the origin at \( (0, 0) \).
- **\( f'(0) = 3 \)**: The slope of the tangent at \( x = 0 \) is 3, meaning the graph is rising steeply at \( x = 0 \).
- **\( f'(1) = 0 \)**: The slope of the tangent at \( x = 1 \) is 0, indicating a horizontal tangent at \( x = 1 \), which suggests a local maximum or minimum.
- **\( f'(2) = -1 \)**: The slope at \( x = 2 \) is negative, meaning the graph is decreasing.

### Evaluating the given options in the image:

1. **First graph (top-left)**:
   - At \( x = 0 \), the graph passes through the origin, and the slope appears steep and positive, which matches \( f(0) = 0 \) and \( f'(0) = 3 \).
   - At \( x = 1 \), there is a horizontal tangent, which satisfies \( f'(1) = 0 \).
   - After \( x = 1 \), the graph starts decreasing, consistent with \( f'(2) = -1 \).

2. **Second graph (top-right)**:
   - At \( x = 0 \), the graph passes through the origin, but the slope seems less steep than 3.
   - At \( x = 1 \), the slope is not clearly horizontal, which does not satisfy \( f'(1) = 0 \).

3. **Third graph (bottom-left)**:
   - At \( x = 0 \), the graph does not pass through the origin, so it doesn't satisfy \( f(0) = 0 \).

4. **Fourth graph (bottom-right)**:
   - The graph seems to meet some conditions, but at \( x = 0 \), the slope appears incorrect for \( f'(0) = 3 \), and the graph doesn't match all conditions.

### Conclusion:
The **first graph (top-left)** is the correct choice. It satisfies all the given conditions for \( f(x) \), including the correct values of the function and its derivatives at the specified points.

Would you like more details, or do you have any questions about this? 

Here are 5 related questions for further exploration:

1. What is the significance of a function's derivative being zero at a point?
2. How can you determine concavity from a graph?
3. What does it mean when the derivative of a function is positive or negative?
4. How can you find local maxima and minima using the first derivative?
5. What is the geometric interpretation of the second derivative?

**Tip**: A function’s first derivative gives the slope of the tangent line, while the second derivative provides information about the curvature (concavity) of the graph.

Sketch the graph of a function f for which f(0) = 0, f'(0) = 3, f'(1) = 0, and f'(2) = -1.

Learn how to sketch the graph of a function with specified derivative conditions, including f(0) = 0, f'(0) = 3, f'(1) = 0, and f'(2) = -1. This problem involves the analysis of a function's behavior using critical points and slopes. Suitable for Grade 12 or college-level students studying calculus.

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