https://r2.erweima.ai/mathbot/image/1722602218004ui7vw96i.jpg

To solve this problem, we need to analyze the given function \( y = f(x) \) to determine where it is increasing, decreasing, and stationary, and then sketch the graph of its derivative \( y = f'(x) \).

1. **Stationary Points**: These are points where the derivative \( f'(x) = 0 \). Looking at the graph, these are the points where the tangent to the curve is horizontal.
   - The graph shows stationary points at approximately \( x = -2.1 \) and \( x = -0.5 \).

2. **Increasing and Decreasing Intervals**:
   - The function \( f(x) \) is increasing when \( f'(x) > 0 \).
   - The function \( f(x) \) is decreasing when \( f'(x) < 0 \).

From the graph:
- For \( x < -2.1 \), the function is decreasing.
- For \( -2.1 < x < -0.5 \), the function is increasing.
- For \( x > -0.5 \), the function is decreasing.

3. **Sketching the Derivative \( y = f'(x) \)**:
   - At the stationary points \( x = -2.1 \) and \( x = -0.5 \), \( f'(x) = 0 \).
   - In the interval \( (-\infty, -2.1) \), the function is decreasing, so \( f'(x) < 0 \).
   - In the interval \( (-2.1, -0.5) \), the function is increasing, so \( f'(x) > 0 \).
   - In the interval \( (-0.5, \infty) \), the function is decreasing again, so \( f'(x) < 0 \).

To illustrate this, let's create the derivative graph:

1. **Draw Points**: Mark the points \( (-2.1, 0) \) and \( (-0.5, 0) \) where the derivative is zero.
2. **Draw Increasing/Decreasing**:
   - For \( x < -2.1 \), \( f'(x) \) should be negative.
   - For \( -2.1 < x < -0.5 \), \( f'(x) \) should be positive.
   - For \( x > -0.5 \), \( f'(x) \) should be negative.

Let's plot this in Python:

```python
import matplotlib.pyplot as plt
import numpy as np

# Given function behavior
x = np.linspace(-5, 3, 400)
y = np.piecewise(x, [x < -2.1, (-2.1 <= x) & (x <= -0.5), x > -0.5], 
                 [lambda x: -1, lambda x: 1, lambda x: -1])

# Derivative points
x_points = [-2.1, -0.5]
y_points = [0, 0]

# Plot
plt.figure(figsize=(10, 6))

# Original function sketch
plt.plot(x, y, 'r-', label='Derivative of f(x)')
plt.scatter(x_points, y_points, color='black')  # Stationary points

plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)

# Annotations
plt.annotate('(-2.1, 0)', (-2.1, 0), textcoords="offset points", xytext=(0,10), ha='center')
plt.annotate('(-0.5, 0)', (-0.5, 0), textcoords="offset points", xytext=(0,10), ha='center')

plt.title('Graph of the Derivative of the Function')
plt.xlabel('x')
plt.ylabel("f'(x)")
plt.legend()
plt.show()
```

The resulting plot will show the correct behavior of the derivative function.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the derivative of a function graphically?
2. What are the conditions for a function to have a stationary point?
3. How does the second derivative help in analyzing the concavity of a function?
4. How do you determine the intervals of concavity for a function?
5. What are inflection points and how are they identified?
6. Can you explain the relationship between a function and its first and second derivatives?
7. How do you use the first derivative test to identify local maxima and minima?
8. What are some common applications of derivatives in real-world problems?

**Tip:** When analyzing the graph of a derivative, remember that the sign of the derivative indicates whether the original function is increasing or decreasing.

Learn how to solve and graph the derivative of a function step-by-step. Understand the concepts of derivatives, stationary points, increasing and decreasing intervals, and sketching derivative graphs with this comprehensive guide. Suitable for Grades 11-12.

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