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The problem involves analyzing a contour plot of a function \( f(x, y) \), where we are asked to identify local maxima and minima, along with the global maximum and minimum.

### Step-by-Step Process:

1. **Understanding the Contour Plot**:
   - The contour lines represent level sets of \( f(x, y) \), where the function takes constant values.
   - The closer the contour lines, the steeper the slope.
   - Peaks (local maxima) are often enclosed by higher-level contours, and valleys (local minima) are enclosed by lower-level contours.

2. **Identifying Critical Points**:
   - Local maxima occur at points enclosed by contour lines with increasing values outward.
   - Local minima occur at points enclosed by contour lines with decreasing values outward.

### Analysis of the Contour Plot:

- **Local Maxima**:
  - There is a local maximum at the top right corner (near \( x = 35 \), \( y = 35 \)) where the contour values increase outward. The highest contour line is labeled 12.
  - Another possible local maximum is at \( x = 10 \), \( y = 30 \), with a surrounding contour value of 11.

- **Local Minima**:
  - In the center of the lower-left part of the plot, near \( x = 20 \), \( y = 20 \), the contour values decrease inward, with a minimum contour value of 6.
  - A second minimum appears near \( x = 30 \), \( y = 10 \), also with values decreasing inward to 6.

- **Global Maximum and Minimum**:
  - **Global Maximum**: The highest value on the plot is at the local maximum where the contour value reaches 12 (near \( x = 35 \), \( y = 35 \)).
  - **Global Minimum**: The lowest contour value is at 6, occurring at two points (near \( x = 20 \), \( y = 20 \), and \( x = 30 \), \( y = 10 \)).

### Answers:
- **Local Maxima**:
  - \( f(10, 30) = 11 \)
  - \( f(35, 35) = 12 \)
  
- **Local Minima**:
  - \( f(20, 20) = 6 \)
  - \( f(30, 10) = 6 \)

- **Global Maximum**: Occurs at \( (35, 35) \) with \( f(35, 35) = 12 \).

- **Global Minimum**: Occurs at \( (20, 20) \) or \( (30, 10) \) with \( f(20, 20) = 6 \) and \( f(30, 10) = 6 \).

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Do you want any more details on how to interpret contour plots or why these points were chosen? 

Here are 5 follow-up questions:
1. What is the significance of contour spacing in a contour plot?
2. How do we determine if a point is a local maximum or minimum using derivatives?
3. Can there be multiple global minima or maxima in a function’s domain?
4. What is the role of second-order partial derivatives in classifying critical points?
5. How can this analysis be extended to three-dimensional surfaces?

**Tip**: Contour lines help visualize multi-variable functions without needing a 3D plot.

The figure below shows contours of f(x, y). Assume the entire domain of f(x, y) is shown. List x- and y-coordinates and the value of the function at any local maximum and local minimum points, and identify which is which. Are any of these local extrema also global extrema on the region shown?

Learn how to analyze contour plots to identify local maxima, minima, and global extrema in a multivariable function. This problem involves analyzing a plot where level curves indicate the function's behavior, suitable for university-level calculus.

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