Based on the graph you provided, let's address each question in turn:

1. For what values of $x$ does the function $f$ have a local (relative) maximum on $[0, 8]$?

   • From the graph, there is a local maximum at $x = 4$ where the curve peaks before descending again.
   • Answer: $x = 4$

2. For what values of $x$ does the function $f$ have a local (relative) minimum on $[0, 8]$?

   • The graph shows a local minimum at $x = 6$, where the curve dips before rising again.
   • Answer: $x = 6$

3. Find the absolute maximum value for the function $f$ on the interval $[0, 8]$.

   • The highest point on the graph is at $x = 0$, where $f(0) = 5$.
   • Answer: The absolute maximum is $5$.

4. Find the absolute minimum value for the function $f$ on the interval $[0, 8]$.

   • The lowest point on the graph is at $x = 6$, where $f(6) = 0$.
   • Answer: The absolute minimum is $0$.

Would you like further details on any of these? Here are five related questions to consider:

1. What is the difference between local and absolute extrema?
2. Can a function have more than one absolute maximum or minimum?
3. How does the derivative of a function relate to its local maxima and minima?
4. Can a function have a local maximum at an endpoint of its interval?
5. What happens if a function has no local extrema on a given interval?

Tip: Always check endpoints of the interval when looking for absolute maximum and minimum values!