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

Step 1: Identifying Local Maxima and Minima

From the graph, it is clear that we are looking for points where the function changes direction, which would indicate relative (local) maxima and minima.

• Relative Maximum occurs when the function changes from increasing to decreasing. Looking at the graph, this happens around $x = 1$.
• Relative Minimum occurs when the function changes from decreasing to increasing. We see this at approximately $x = 4$ and $x = 7$.

Step 2: Answering Questions

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

   • Answer: $x = 1$

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

   • Answer: $x = 4, 7$

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

   • The absolute maximum is the highest point on the graph over the interval. From the graph, the highest value appears at $x = 0$, where $f(x) \approx 2$.
   • Answer: $2$

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

   • The absolute minimum is the lowest point on the graph. This occurs near $x = 5$, where $f(x) \approx -1.5$.
   • Answer: $-1.5$

Would you like a more detailed explanation of how to identify these points, or do you have any further questions?

Related Questions

1. What is the difference between relative and absolute extrema?
2. How can you use the derivative of a function to find critical points?
3. Can a function have more than one absolute maximum or minimum?
4. Why is the endpoint included when determining the absolute maximum or minimum on a closed interval?
5. How does the second derivative help confirm whether a point is a maximum or minimum?

Tip: To confirm whether a point is a maximum or minimum, check if the slope of the tangent (first derivative) changes sign at that point.