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The question is asking about the local extreme points for the function \( f(x) \), based on the graph of its derivative \( f'(x) \). From the graph, we can observe that:

- Local maxima occur where \( f'(x) = 0 \) and changes from positive to negative (i.e., the graph of \( f'(x) \) crosses the x-axis from above to below).
- Local minima occur where \( f'(x) = 0 \) and changes from negative to positive (i.e., the graph of \( f'(x) \) crosses the x-axis from below to above).

### Analysis:
- At point \( A \), \( f'(x) = 0 \) and changes from negative to positive, indicating a local minimum.
- At point \( B \), \( f'(x) = 0 \) and changes from positive to negative, indicating a local maximum.
- At point \( D \), \( f'(x) = 0 \) and changes from negative to positive, indicating another local minimum.

Thus, the correct local extreme points for \( f(x) \) are:
- Local minimum at \( A \) and \( D \).
- Local maximum at \( B \).

### Answer:
The correct option is:
- **\( f(x) \) has a local minimum at \( x = A \) and \( x = D \); and local maximum at \( x = B \).**

Would you like further details or clarification?

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Here are five related questions you might consider:
1. What happens to \( f(x) \) when \( f'(x) \) is positive or negative?
2. How can you determine inflection points from the second derivative \( f''(x) \)?
3. What does the concavity of \( f(x) \) indicate about the sign of \( f'(x) \)?
4. How can you identify global extrema on a closed interval?
5. What role does the second derivative test play in determining extreme points?

**Tip:** Always check the sign of the derivative around critical points to confirm whether they are maxima, minima, or inflection points.

The graph below belongs to f'(x), the first derivative of a polynomial f(x). List all local extreme points for f(x).

Learn how to identify local minima and maxima by analyzing the graph of the first derivative f'(x) of a polynomial function. This problem covers key calculus concepts such as critical points and the First Derivative Test, suitable for students in Grades 11-12 or early college.

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