Use the graph of the function f given below to answer the questions.
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Part 1
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A coordinate system has a horizontal x-axis labeled from negative 14 to 14 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A curve rises from left to right to a plotted point at (negative 9, 9), falls from left to right to a plotted point at (negative 3, negative 9), rises from left to right, passing through the origin, to a plotted point at (3, 9), falls from left to right to a plotted point at (9, negative 9), and then rises from left to right.
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Question content area right
Part 1
List the values of x at which f has a local minimum. Select the correct choice below and fill in any answer boxes within your choice.
A.
xequals

enter your response here
​(Type an integer. Use a comma to separate answers as​ needed.)
B.
There are no local minima.

The graph you described shows a function \( f(x) \) that has a periodic-like behavior with alternating rises and falls.

### Identifying Local Minima:
A local minimum occurs where the function changes from decreasing to increasing. From the description of the graph, we can identify the following key points:

- The function **falls** from \( (-9, 9) \) to \( (-3, -9) \), meaning at \( x = -3 \), the function reaches a low point (local minimum).
- The function **falls** again from \( (3, 9) \) to \( (9, -9) \), so at \( x = 9 \), the function also reaches another low point (local minimum).

Thus, the local minima occur at \( x = -3 \) and \( x = 9 \).

### Answer:
The values of \( x \) at which \( f(x) \) has local minima are:

\[
x = -3, 9
\]

### Would you like a more detailed explanation of how to find local minima or have further questions?

### Related Questions:
1. What are the values of \( x \) at which \( f(x) \) has local maxima?
2. What is the behavior of the function as \( x \to -\infty \) and \( x \to \infty \)?
3. Can you identify points of inflection on this graph?
4. How do we distinguish between a local minimum and a global minimum?
5. How can you use the first and second derivative tests to find minima?

#### Tip:
When identifying local extrema, always check where the function changes direction (increasing to decreasing for maxima, decreasing to increasing for minima).

Use the graph of the function f given below to answer the questions.
List the values of x at which f has a local minimum.

Analyze the graph of a function to determine the x-values where local minima occur. This problem focuses on the behavior of functions and how to identify local extrema by observing graph trends. Suitable for students in Grades 10-12.

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