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To determine where \( f \) has local maximum points using the graph of its derivative \( f'(x) \), follow these steps:

### Understanding the Relationship:
1. **Local Maximums** occur where:
   - \( f'(x) = 0 \) (critical points).
   - The derivative changes from **positive to negative** at these points.
   
   On the graph, \( f'(x) = 0 \) where the curve crosses the x-axis. We are looking for places where the graph crosses the x-axis **from above** (positive) to below (negative), indicating a local maximum for \( f(x) \).

### Steps to Solve:
1. **Find the Critical Points**:
   Look for the x-values where the graph of \( f'(x) \) crosses the x-axis. These points are approximately:
   - \( x \approx -3.5 \)
   - \( x = 0 \)
   - \( x = 5 \)

2. **Determine the Sign Change**:
   Check the behavior of \( f'(x) \) around these points:
   - Around \( x = -3.5 \), the graph moves from positive to negative. This indicates a **local maximum** for \( f \).
   - At \( x = 0 \), the graph moves from negative to positive, indicating a **local minimum** for \( f \).
   - At \( x = 5 \), the graph moves from positive to negative, which indicates another **local maximum** for \( f \).

Thus, the local maxima of \( f(x) \) occur approximately at \( x \approx -3.5 \) and \( x = 5 \).

In particular, the x-value \( x = 5 \) is a local maximum because \( f'(x) \) changes from positive to negative at this point.

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

Here are 5 related questions to explore further:
1. How do we classify points of inflection using a derivative graph?
2. Why is it important to check the sign change of the derivative at critical points?
3. How do local minima differ from local maxima in terms of the derivative?
4. What happens if \( f'(x) \) doesn't change sign at a critical point?
5. How can we determine if there are any global maxima or minima from a derivative graph?

**Tip:** Always check the behavior of the derivative on either side of a critical point to confirm whether it's a local maximum or minimum.

Learn how to find the local maxima of a function based on its derivative graph, focusing on identifying critical points and analyzing sign changes in the derivative. In this case, we solve for x = 5.

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