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To sketch a possible graph of \( f \) based on the given information, let's analyze the details provided in the problem.

### Given Information
1. **Function Values**:
   - \( f(-1) = 3 \)
   - \( f(2) = -4 \)
   
   These are two points on the graph of \( f \): \( (-1, 3) \) and \( (2, -4) \).

2. **Sign of the Derivative**:
   - The derivative \( f'(x) \) changes sign at \( x = -1 \) and \( x = 2 \).
   - The sign of \( f'(x) \) in different intervals is given:
     - For \( x < -1 \): \( f'(x) > 0 \) (positive slope, \( f \) is increasing).
     - For \( -1 < x < 2 \): \( f'(x) < 0 \) (negative slope, \( f \) is decreasing).
     - For \( x > 2 \): \( f'(x) > 0 \) (positive slope, \( f \) is increasing).

### Interpretation for Sketching \( f \)
Using the information about \( f(-1) \), \( f(2) \), and the sign changes in \( f'(x) \):
- **At \( x = -1 \)**: The derivative \( f'(x) \) changes from positive to negative, indicating a local maximum at \( x = -1 \). Therefore, the point \( (-1, 3) \) is a local maximum.
- **At \( x = 2 \)**: The derivative \( f'(x) \) changes from negative to positive, indicating a local minimum at \( x = 2 \). Therefore, the point \( (2, -4) \) is a local minimum.

### Steps to Sketch \( f \)
1. **Draw Points**: Plot the points \( (-1, 3) \) and \( (2, -4) \) on a coordinate system.
2. **Behavior of \( f(x) \) Between Points**:
   - For \( x < -1 \): The function is increasing, approaching the local maximum at \( (-1, 3) \).
   - For \( -1 < x < 2 \): The function is decreasing from the maximum at \( (-1, 3) \) to the minimum at \( (2, -4) \).
   - For \( x > 2 \): The function is increasing from the minimum at \( (2, -4) \).

3. **Sketch**: Based on the increasing and decreasing behavior in each interval, sketch a smooth curve that:
   - Starts by increasing for \( x < -1 \),
   - Peaks at \( (-1, 3) \),
   - Decreases to \( (2, -4) \),
   - Then starts increasing again for \( x > 2 \).

This sketch will show a graph that reflects the specified behavior of \( f(x) \) and \( f'(x) \) over the intervals.

Would you like a more detailed explanation, or do you have any questions?

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Here are 5 related questions to deepen your understanding:
1. What does it mean for a function to have a local maximum or minimum?
2. How can we determine if a point is a maximum or minimum based on the derivative?
3. How would the graph change if \( f'(x) \) did not change signs at \( x = -1 \) or \( x = 2 \)?
4. What additional information could we use to further refine the graph of \( f(x) \)?
5. How can the second derivative test help in identifying concavity near critical points?

**Tip**: When sketching the graph of a function based on derivative information, focus on identifying intervals of increase and decrease, as well as the points where these intervals change.

Sketch the graph of a differentiable function f with the following properties:

f(-1) = 3 and f(2) = -4.
The zeros and signs of f'(x) are indicated on the number line:

x: -1  2
f'(x): +  0  -  0  +

Sketch the graph of f.

Learn how to sketch a graph of a function f based on given function values and the sign changes in its derivative. This example covers essential concepts in calculus, including critical points, increasing and decreasing behavior, and using the first derivative test. Suitable for students in Grades 11-12.

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