https://file.aiquickdraw.com/mathbot/file/1730300434692265615tt.jpg

From the provided image, we need to determine the relative minimum and maximum points based on the graph.

Let's break down each part of the question:

### (a) For what value(s) of \( x \) does the function obtain a relative minimum?
Observing the graph, it appears that there is a minimum point at \( x = -1 \) where the graph dips down to the lowest local value in that vicinity. So:

- **Answer:** The function has a relative minimum at \( x = -1 \).

### (b) Find the relative minimum value.
The relative minimum value corresponds to the \( y \)-value at \( x = -1 \). According to the graph, the point at \( x = -1 \) has \( y = -9 \).

- **Answer:** The relative minimum value is \( -9 \).

### (c) For what value(s) of \( x \) does the function obtain a relative maximum?
Looking at the graph, a relative maximum appears to occur at \( x = -7 \), where the function reaches a peak locally.

- **Answer:** The function has a relative maximum at \( x = -7 \).

### (d) Find the relative maximum value.
The relative maximum value corresponds to the \( y \)-value at \( x = -7 \). From the graph, this point appears to have \( y = 7 \).

- **Answer:** The relative maximum value is \( 7 \).

If you'd like more details or have any questions about this solution, feel free to ask!

Here are some related questions to explore:
1. How do we determine relative minima and maxima from the derivative of a function?
2. What is the difference between absolute and relative extrema?
3. Can a function have more than one relative minimum or maximum?
4. What role does the second derivative play in identifying minima and maxima?
5. How can we estimate values from a graph if we don't have precise coordinates?

**Tip:** When analyzing a graph, look for points where the slope changes from positive to negative (for maximum) or from negative to positive (for minimum).

For what value(s) of x does the function obtain a relative minimum? Find the relative minimum value. For what value(s) of x does the function obtain a relative maximum? Find the relative maximum value.

Analyze a graph to find relative minimum and maximum points. Learn how to determine the x-values where the function obtains these values and calculate the corresponding y-values. This problem is suitable for high school calculus students in Grades 11-12.

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