Final Answer:
The correct interval for the graphed function with a local minimum of 0 is option B. (-2, 0).
Step-by-step explanation:
In order to determine the interval with a local minimum of 0, we need to analyze the behavior of the function around critical points where the derivative is equal to zero. A local minimum occurs when the function transitions from decreasing to increasing. In this case, the interval (-2, 0) is the correct choice.
Firstly, we identify critical points by setting the derivative of the function equal to zero:
[f'(x) = 0.]
After finding these critical points, we analyze the sign of the derivative in the intervals determined by these points. In the interval (-3, -2), if \(f'(x) > 0\), the function is increasing; if \(f'(x) < 0\), the function is decreasing. Similarly, in the interval (-2, 0), a negative derivative indicates a decreasing function, while a positive derivative indicates an increasing function.
Next, by checking the behavior of the function in these intervals, we observe that in (-2, 0), the function transitions from decreasing to increasing, indicating a local minimum. Therefore, the correct choice is (-2, 0) for the interval with a local minimum of 0.