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?

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.