Let's analyze the graph and answer the questions based on it.

Observations from the Graph:

• The function is plotted over the interval $[0, 8]$.
• We can identify the local maxima, minima, and endpoints from the curve's behavior.

Answering the Questions:

1. Local Maximum on (0, 8): A local maximum occurs where the graph changes from increasing to decreasing. Based on the graph, the local maximum appears at $x = 2$, where the function peaks before starting to decrease.

   Answer: $x = 2$

2. Local Minimum on (0, 8): A local minimum occurs where the graph changes from decreasing to increasing. The function reaches local minima at:

   • Around $x = 5$, as the graph dips and then rises again.

   Answer: $x = 5$

3. Absolute Maximum on [0, 8]: The highest point on the graph across the interval $[0, 8]$ is around $x = 0$, where the function value is approximately 5.

   Answer: Absolute maximum $\approx 5$

4. Absolute Minimum on [0, 8]: The lowest point on the graph across the interval $[0, 8]$ occurs around $x = 7$, where the function value is approximately 0.

   Answer: Absolute minimum $\approx 0$

Would you like further details on any specific part of this problem? Let me know if you have any questions!

Related Questions:

1. What is the difference between local and absolute extrema?
2. How can you find extrema using calculus?
3. How can the endpoints of a function affect the absolute extrema?
4. What is the first derivative test for determining local maxima and minima?
5. What is the role of concavity in identifying local extrema?

Tip:

When identifying extrema, remember that local extrema depend on the behavior of the function in small neighborhoods, while absolute extrema depend on the entire interval being considered.