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The figure depicts the graph of a function f(x). Solve the inequality f'(x) <_ 0.

To solve the inequality \( f'(x) \leq 0 \) using the graph of the function \( f(x) \), we need to identify the intervals on the \( x \)-axis where the slope of the function is non-positive (i.e., where the graph of \( f(x) \) is either decreasing or has a zero slope).

### Steps to Solve:

1. **Identify where \( f'(x) = 0 \):**
   - \( f'(x) = 0 \) at points where the graph has horizontal tangents, which occur at the local maxima, minima, or inflection points.
   - By examining the graph, we see that horizontal tangents occur at the following points:
     - \( x \approx 0.5 \) (local maximum)
     - \( x \approx 3.5 \) (local minimum)

2. **Identify where \( f'(x) < 0 \):**
   - \( f'(x) < 0 \) where the graph is decreasing. This happens between the local maximum and minimum.
   - The function is decreasing (negative slope) between:
     - \( x = 0.5 \) and \( x = 3.5 \).

3. **Combine the information:**
   - \( f'(x) \leq 0 \) when the function is either decreasing or flat. Based on the graph:
     - The inequality holds from \( x \approx 0.5 \) to \( x \approx 3.5 \).

### Final Answer:
The solution to the inequality \( f'(x) \leq 0 \) is the interval:
\[
0.5 \leq x \leq 3.5
\]

Would you like more details or have any questions on this? Here are some additional questions to explore:

1. What does \( f'(x) > 0 \) represent on the graph?
2. How can you identify points of inflection using the graph?
3. How would the graph change if the function had an additional local minimum?
4. Can you determine the concavity of the function in different intervals based on the graph?
5. What can you infer about the second derivative \( f''(x) \) at the points where \( f'(x) = 0 \)?

**Tip:** When analyzing the slope of a function graphically, always consider the intervals of increase and decrease and watch for turning points where the slope changes sign.

Learn how to solve the inequality f'(x) <= 0 using graphical analysis of the function f(x). This problem covers calculus concepts, including derivatives and graphical interpretation. Suitable for advanced high school students.

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