https://file.aiquickdraw.com/mathbot/file/17277952632980nob4m3l.jpg

the  x-axis and y-axis  are going by ten. sovle 

Let's analyze the graph of the function step by step and solve the questions.

### Part (a) Local Maxima and Minima:

We need to identify the local maxima and minima by estimating the coordinates of the points where the graph reaches a peak (max) or a trough (min).

1. **First Local Maximum (smaller x-value)**:  
   Looking at the graph, the first local maximum occurs near \( x = -5 \), and the corresponding \( y \)-value is approximately 10.  
   So, the first local maximum is \( (x, y) = (-5, 10) \).

2. **Second Local Maximum (larger x-value)**:  
   The second local maximum occurs near \( x = 5 \), and the \( y \)-value is approximately 5.  
   So, the second local maximum is \( (x, y) = (5, 5) \).

3. **Local Minimum**:  
   The local minimum occurs near \( x = 0 \), and the corresponding \( y \)-value is approximately \(-10\).  
   So, the local minimum is \( (x, y) = (0, -10) \).

### Part (b) Intervals of Increasing and Decreasing:

- **Increasing intervals**:  
   The function is increasing where the slope of the graph is positive. This occurs between:
   1. \( (-\infty, -5) \) 
   2. \( (0, 5) \)

- **Decreasing intervals**:  
   The function is decreasing where the slope of the graph is negative. This occurs between:
   1. \( (-5, 0) \)
   2. \( (5, \infty) \)

### Final Answers:

**(a)** Local extrema:
- First local maximum: \( (-5, 10) \)
- Second local maximum: \( (5, 5) \)
- Local minimum: \( (0, -10) \)

**(b)** Intervals of increasing and decreasing:
- Increasing: \( (-\infty, -5) \cup (0, 5) \)
- Decreasing: \( (-5, 0) \cup (5, \infty) \)

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Do you have any questions or need further clarification on any part of the solution?

Here are 5 related questions to expand your understanding:
1. How would you find the global maximum of the function from the graph?
2. What is the importance of identifying local maxima and minima in real-world scenarios?
3. How does calculus help in determining the exact points of local maxima and minima?
4. Can a function have multiple local maxima but no global maximum?
5. What is the difference between local and absolute extrema?

**Tip:** When identifying local maxima and minima, always observe where the slope of the graph changes from positive to negative (maxima) or from negative to positive (minima).

The graph of a function f is given. Use the graph to estimate all the local maximum and minimum values of the function and the value of x at which each occurs. Additionally, determine the intervals on which the function is increasing and decreasing.

Learn how to analyze the local maximum and minimum values of a function based on its graph. This problem covers key concepts in graph analysis, such as identifying local extrema and understanding intervals of increase and decrease. Suitable for students in Grades 11-12.

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