The image you've uploaded contains a question related to the behavior of a function and its derivative at a given point. Here's a summary of the conditions:

1. Maximum at the point $x_0$: The derivative changes sign from negative to positive at this point.
2. Minimum at the point $x_0$: The derivative does not change sign at this point.
3. Extremum at the point $x_0$: The derivative does not exist at this point.

Do you need any further clarification on this topic or additional examples?

Here are some questions to explore further:

1. What are the conditions for a function to have a local maximum or minimum?
2. How do second derivatives help in classifying critical points?
3. What happens when the first derivative does not exist at a point?
4. How do inflection points relate to the behavior of the derivative?
5. Can a function have a local extremum without a critical point?

Tip: The sign of the first derivative gives important clues about the behavior of the function at a point: positive means increasing, negative means decreasing, and zero can indicate a maximum, minimum, or inflection point.