From the graph you've shared, we can analyze the function $f(x)$ and its critical points. A critical point occurs where the derivative of the function is zero or undefined, typically at peaks, troughs, or points of inflection.

Here's how to address each question:

1. How many critical points does $f$ have? Critical points are located where the slope of the curve becomes zero (horizontal tangent). From the graph, it appears that the function has critical points at:

   • $B$, $D$, $F$, $H$, and $J$.

2. Which points are critical points? Based on the labeled graph, the critical points are: B, D, F, H, J.

3. At which points does $f$ have a local maximum? A local maximum occurs where the function changes from increasing to decreasing (a peak). Looking at the graph, the local maxima are at:

   • B and H.

4. At which points does $f$ have a local minimum? A local minimum occurs where the function changes from decreasing to increasing (a trough). The local minima are at:

   • D and J.

Let me know if you need further clarification!

Expanding questions:

1. What is the behavior of the function between the critical points?
2. How can you determine if a critical point is a maximum, minimum, or inflection point?
3. Can you find the intervals where the function is increasing or decreasing?
4. How would the second derivative help in classifying the critical points?
5. Could this function have any inflection points based on the graph?

Tip:

To confirm whether a critical point is a maximum or minimum, examine the sign of the derivative before and after the point.