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To find the general appearance of \( f'(x) \) based on the given graph of \( f(x) \), we need to interpret how the slopes of the original function change across the curve. Here’s the breakdown:

1. **Where \( f'(x) = 0 \) (Critical Points)**:
   The derivative \( f'(x) \) is zero where the function \( f(x) \) has horizontal tangents (i.e., at local maxima, minima, or points of inflection where the slope is zero). From the graph:
   - At \( x = -2 \), the slope is zero (local maximum).
   - At \( x = 0 \), the slope is zero (local minimum).

2. **Intervals Where \( f'(x) > 0 \) (Positive Slope)**:
   - Between \( x = -\infty \) and \( x = -2 \), the function is increasing, meaning \( f'(x) > 0 \).
   - Between \( x = 0 \) and \( x = \infty \), the function is increasing again, so \( f'(x) > 0 \).

3. **Intervals Where \( f'(x) < 0 \) (Negative Slope)**:
   - Between \( x = -2 \) and \( x = 0 \), the function is decreasing, meaning \( f'(x) < 0 \).

### Approximate Shape of \( f'(x) \):
- At \( x = -2 \) and \( x = 0 \), there will be x-intercepts (because the slope is zero at those points).
- Between \( -2 \) and \( 0 \), the derivative will be negative.
- To the left of \( x = -2 \) and to the right of \( x = 0 \), the derivative will be positive.

Thus, \( f'(x) \) should resemble a function that has roots at \( x = -2 \) and \( x = 0 \) and is positive on the intervals \( (-\infty, -2) \) and \( (0, \infty) \), while being negative between \( -2 \) and \( 0 \). This suggests that \( f'(x) \) is likely a parabola-shaped curve that opens upwards, possibly something similar to a quadratic function.

Would you like to see more details on this, or have any further questions?

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**Follow-up Questions**:
1. What are the inflection points of \( f(x) \), and how do they affect \( f'(x) \)?
2. How can we determine the concavity of \( f(x) \) based on \( f'(x) \)?
3. Can you sketch \( f'(x) \) based on this graph of \( f(x) \)?
4. How do critical points of \( f(x) \) relate to the second derivative, \( f''(x) \)?
5. What would the second derivative, \( f''(x) \), look like based on this graph of \( f(x) \)?

**Tip**: Remember that the sign of the first derivative \( f'(x) \) indicates whether the function is increasing or decreasing.

What does f'(x) look like if this is f(x)?

Understand how to determine the shape of f'(x) based on the graph of f(x). This solution covers key concepts in calculus, such as critical points, increasing and decreasing intervals, and interpreting slope changes.

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