Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.7 Optimization in Several Variables - Preliminary Questions - Page 821: 2

Answer

1. First graph of Figure 17 The point is a saddle point. 2. Second graph of Figure 17 The point is not a local minima nor local maxima. It is neither a saddle point. 3. Third graph of Figure 17 The left point is a local minimum. The right point is a local maximum.

Work Step by Step

1. First graph of Figure 17 If we walk in the directions $ + {\bf{i}}$ or $ - {\bf{i}}$, it will take us downhill. If we walk in the directions $ + {\bf{j}}$ or $ - {\bf{j}}$, it will take us uphill. Therefore, we conclude that it is a saddle point. 2. Second graph of Figure 17 If we walk in these directions: $ + {\bf{i}}$, it will take us uphill $ - {\bf{i}}$, it will take us downhill $ + {\bf{j}}$ or $ - {\bf{j}}$, it is neither downhill nor uphill Therefore, we conclude that it is not a local minima nor local maxima. It is neither a saddle point. 3. Third graph of Figure 17 The left point is a local minimum, because no matter which direction we walk, it will take us uphill. The right point is a local maximum, because no matter which direction we walk, it will take us downhill.
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