Let's analyze the contour diagram to identify the **local maxima**, **local minima**, and **saddle points**. ### **Step-by-Step Solution** #### **Step 1: Understanding Contour Diagrams** - **Contours** represent level curves for a function \( f(x, y) \). - **Local maxima** are points surrounded by decreasing values (the highest point in their neighborhood). - **Local minima** are points surrounded by increasing values (the lowest point in their neighborhood). - **Saddle points** are points where the contours indicate a transition from increasing to decreasing value in different directions (neither maxima nor minima). #### **Step 2: Identify Local Maxima** - Look for the smallest closed contour with **highest value** at the center. - On the diagram: - **Top right**: At the center of the smallest circle with the highest label (value 16), the values decrease as we move away. - **Local maximum at** approximately \((2.5, 2)\). #### **Step 3: Identify Local Minima** - Look for the smallest closed contour with **lowest value** at the center. - On the diagram: - **Bottom right**: At the center of the smallest circle with the lowest label (value 0), the values increase as we move away. - **Local minimum at** approximately \((2, -1.5)\). #### **Step 4: Identify Saddle Points** - Saddle points occur where contours pinch together, indicating a change from increasing to decreasing in different directions. - On the diagram: - **Between the two top contours** (left-center) and the bottom right minimum, a "pinched" contour shape appears. - **Saddle point** at approximately \((0.5, 0)\). - **Between the bottom minimum and the left contour**, another pinch is visible. - **Saddle point** at approximately \((-1, -0.5)\). #### **Step 5: Check for Other Extrema** - **Left Center**: There is a small closed contour just left of the center at around \((-1, 1)\) with value increasing outward. - **Local minimum** at \((-1, 1)\). - **Center**: There is a closed contour at the center near \((1, 0.5)\) with value increasing outward. - **Local minimum** at \((1, 0.5)\). --- ### **Final Answer:** **Local Maxima:** - \((2.5, 2)\) **Local Minima:** - \((2, -1.5)\) - \((-1, 1)\) - \((1, 0.5)\) **Saddle Points:** - \((0.5, 0)\) - \((-1, -0.5)\) --- #### **Explanation:** - Local maxima: Center of closed contours with highest value. - Local minima: Center of closed contours with lowest value. - Saddle points: At "pinched" or crossing contours. If you need a drawing or marking on the diagram, let me know!

Certainly! To provide a detailed calculation and explanation for identifying the local maxima, minima, and saddle points from the contour diagram, we'll proceed systematically. --- ### **Step 1: Understanding the Contour Diagram** The contour diagram shows level curves of a function \(f(x, y)\). The labels on the contours indicate the function's value at those curves. The critical points (maxima, minima, saddle points) are typically located at characteristic features such as: - Centers of closed contours (maxima or minima). - Points where contours pinch or intersect (saddle points). --- ### **Step 2: Identifying Critical Points** **A. Local Maxima** - Look for points **inside the highest-value contours** that form a closed loop. - Typically, the **innermost contour** with the **highest value** indicates a local maximum. **Observation:** - The innermost contour near the top right has the highest value label: **16**. - The contour surrounds a point approximately at **(2.5, 2)**. - The values decrease as we move outward, indicating this is a **local maximum**. --- **B. Local Minima** - Look for points **inside the lowest-value contours** that form a closed loop. - The innermost contour with the **lowest value** indicates a local minimum. **Observation:** - Near the bottom right, there's a small closed contour with label **0**. - This contour surrounds approximately **(2, -1.5)**. - Values increase outward, indicating a **local minimum** at this point. --- **C. Saddle Points** - Saddle points are located where contours **pinch or cross**; at these points, the function changes from increasing to decreasing in different directions. - Usually, these points are **not at the center of closed contours** but where contours appear to meet or pinch. **Observation:** - Near **(0.5, 0)**, the contours **pinch together**, indicating a saddle point. - Similarly, at approximately **(-1, -0.5)**, there is evidence of a contour pinch or crossing. --- ### **Step 3: Summarize the Critical Points** | Type | Approximate Coordinates | Explanation | |------------------|---------------------------|--------------------------------------------------------------| | **Local Maxima** | **(2.5, 2)** | Center of highest contour (value 16) | | **Local Minima** | **(2, -1.5)** | Center of lowest contour (value 0) | | **Local Minima** | **(-1, 1)** | Small closed contour with increasing values outward | | **Local Minima** | **(1, 0.5)** | Another small closed contour with increasing values outward | | **Saddle Points**| **(0.5, 0)** | Contour pinch point, indicating a saddle | | **Saddle Points**| **(-1, -0.5)** | Another pinch point in the contour map | --- ### **Step 4: Additional Calculations (if needed)** Since the question is based on the contour diagram and labels, the key calculations involve: - **Evaluating the function \(f(x, y)\) at these points**: Given the diagram's labels, the function's values at these points are inferred from the contours. - **Confirming nature of the points**: - For maxima/minima: Check if the point is at the center of a closed contour with the highest/lowest label. - For saddle points: Confirm the contour shape indicates a transition line where the gradient changes direction. --- ### **Final Summary:** | **Type** | **Coordinates** | **Function Value (from contours)** | **Reasoning** | |---------------------|-----------------|-----------------------------------|-----------------------------------------------------------| | **Local Maximum** | **(2.5, 2)** | **16** | Center of the highest closed contour | | **Local Minima** | **(2, -1.5)** | **0** | Center of the lowest closed contour | | **Saddle Point** | **(0.5, 0)** | Approximate value between 8 and 10 | Contour pinch point indicating a change in the gradient | | **Local Minima** | **(-1, 1)** | Value around 4-6 (from contours) | Closed contour with increasing values outward | | **Saddle Point** | **(-1, -0.5)** | Value around 4-6 (from contours) | Pinch or crossing of contours | --- ### **Note:** - Exact numerical verification would require the explicit function or coordinate grid, but based on the contour labels and shapes, these are the best estimates. - The key is recognizing that **closed contours** with extremal labels indicate maxima or minima, while **pinched or crossing contours** indicate saddle points. --- **If you'd like, I can help you plot these points or interpret specific contour labels further!**