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The contour diagram for z= f(x.y) is shown below. Identify all the critical points off. Also explain the difference between a local extrema( local max and local min) and a saddle point graphically. -2.50 0,5

The contour diagram for z= f(x.y) is shown below. Identify all the critical points off. Also explain the difference between a local extrema( local max and local min) and a saddle point graphically. -2.50 0,5

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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Sure! Here's the transcription and explanation suitable for an educational website: --- **Contour Diagram Analysis** 3. The contour diagram for \( z = f(x, y) \) is shown below. Identify all the critical points. Also, explain the difference between local extrema (local maximum and local minimum) and a saddle point graphically. **Diagram Explanation:** The contour diagram represents level curves for the function \( z = f(x, y) \). These curves are essentially cross-sections of the 3D surface, showing where the function has constant values. Each contour line corresponds to a different value of \( z \). **Key Features of the Diagram:** 1. **Contour Lines**: The diagram consists of several closed loops and intersecting curves. The intervals between lines suggest changes in the function’s value. 2. **Critical Points Identification**: - **Local Extrema**: These occur where the contour lines form closed loops. - Local maxima are identified by the innermost closed loops representing higher \( z \) values than their surroundings. - Local minima are identified by loops representing lower \( z \) values than their surroundings. - **Saddle Points**: These exist where contour lines intersect or form an "X" shape. This indicates that the point is neither a maximum nor a minimum (i.e., the slope changes direction). 3. **Graphical Representation**: - **Local Maximum**: Enclosed contours with lines decreasing outward from the center, suggesting a peak. - **Local Minimum**: Enclosed contours with lines increasing outward, indicating a trough. - **Saddle Point**: Intersecting contours with lines on either side indicating opposite slopes, suggesting a pass or col (a point of equilibrium that is not an extremum). Understanding these critical points on the contour diagram helps in visualizing the topographical nature of the function \( z = f(x, y) \) and is fundamental in calculus to determine the behavior of multivariable functions.
Transcribed Image Text:Sure! Here's the transcription and explanation suitable for an educational website: --- **Contour Diagram Analysis** 3. The contour diagram for \( z = f(x, y) \) is shown below. Identify all the critical points. Also, explain the difference between local extrema (local maximum and local minimum) and a saddle point graphically. **Diagram Explanation:** The contour diagram represents level curves for the function \( z = f(x, y) \). These curves are essentially cross-sections of the 3D surface, showing where the function has constant values. Each contour line corresponds to a different value of \( z \). **Key Features of the Diagram:** 1. **Contour Lines**: The diagram consists of several closed loops and intersecting curves. The intervals between lines suggest changes in the function’s value. 2. **Critical Points Identification**: - **Local Extrema**: These occur where the contour lines form closed loops. - Local maxima are identified by the innermost closed loops representing higher \( z \) values than their surroundings. - Local minima are identified by loops representing lower \( z \) values than their surroundings. - **Saddle Points**: These exist where contour lines intersect or form an "X" shape. This indicates that the point is neither a maximum nor a minimum (i.e., the slope changes direction). 3. **Graphical Representation**: - **Local Maximum**: Enclosed contours with lines decreasing outward from the center, suggesting a peak. - **Local Minimum**: Enclosed contours with lines increasing outward, indicating a trough. - **Saddle Point**: Intersecting contours with lines on either side indicating opposite slopes, suggesting a pass or col (a point of equilibrium that is not an extremum). Understanding these critical points on the contour diagram helps in visualizing the topographical nature of the function \( z = f(x, y) \) and is fundamental in calculus to determine the behavior of multivariable functions.
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