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Use the contour map to determine whether the critical points A, B, C, D are local minima, local maxima, or saddle points. 2- 0- -2+ Point A is a Point B is a Point C is a Point D is a D 0 *C

Use the contour map to determine whether the critical points A, B, C, D are local minima, local maxima, or saddle points. 2- 0- -2+ Point A is a Point B is a Point C is a Point D is a D 0 *C

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Exploring Critical Points Using a Contour Map The task is to analyze the contour map to determine the nature of the critical points labeled \(A\), \(B\), \(C\), and \(D\): specifically, whether they are local minima, local maxima, or saddle points. #### Description of the Contour Map: - **Contours Representing Levels:** The contour lines illustrate different level curves of function \(f(x, y)\). Each line corresponds to a constant function value. - **Point Labels:** Points labeled as \(A\), \(B\), \(C\), and \(D\), are marked on these contours. - **Value Annotation:** Several contour lines are marked with function values, ranging from \(-3\) to \(1\). #### Interpretation of Points: - **Point \(A\):** - Surrounded by higher value contours, indicating a depression. - Likely a local minimum. - **Point \(B\):** - Surrounded by a mix of higher and lower value contours, suggesting instability in directions. - Likely a saddle point. - **Point \(C\):** - Encircled by lower value contours. - Likely a local maximum. - **Point \(D\):** - Similar situation to point \(B\) with mixed contour values surrounding it. - Likely a saddle point. #### Interactive Elements: - Dropdown menus are available to classify each point as a “local minimum,” “local maximum,” or “saddle point.” This educational tool helps in understanding the topological nature of critical points using visual contour analysis.
Transcribed Image Text:### Exploring Critical Points Using a Contour Map The task is to analyze the contour map to determine the nature of the critical points labeled \(A\), \(B\), \(C\), and \(D\): specifically, whether they are local minima, local maxima, or saddle points. #### Description of the Contour Map: - **Contours Representing Levels:** The contour lines illustrate different level curves of function \(f(x, y)\). Each line corresponds to a constant function value. - **Point Labels:** Points labeled as \(A\), \(B\), \(C\), and \(D\), are marked on these contours. - **Value Annotation:** Several contour lines are marked with function values, ranging from \(-3\) to \(1\). #### Interpretation of Points: - **Point \(A\):** - Surrounded by higher value contours, indicating a depression. - Likely a local minimum. - **Point \(B\):** - Surrounded by a mix of higher and lower value contours, suggesting instability in directions. - Likely a saddle point. - **Point \(C\):** - Encircled by lower value contours. - Likely a local maximum. - **Point \(D\):** - Similar situation to point \(B\) with mixed contour values surrounding it. - Likely a saddle point. #### Interactive Elements: - Dropdown menus are available to classify each point as a “local minimum,” “local maximum,” or “saddle point.” This educational tool helps in understanding the topological nature of critical points using visual contour analysis.
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