Answered: 1 f (x, y) = x² + y²x- x 2

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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Related questions

Question

Determine the location of all extremes and show what type the extremes

Expert Solution

Step 1

The extremum values of the functions can be calculated using the second derivative test. For that first find the critical points of the function.

If the first and second-order derivatives exist for the function $f (x, y)$ and let $(x_{0}, y_{0})$ be some points on the disk such that $f_{x} (x_{0}, y_{0}) = 0$ and $f_{y} (x_{0}, y_{0}) = 0$ . Then, $D = f_{x x} (x_{0}, y_{0}) f_{y y} (x_{0}, y_{0}) - {(f_{x y} (x_{0}, y_{0}))}^{2}$ .

If $D > 0$ and $f_{x x} (x_{0}, y_{0}) > 0$ , then f has a local minimum at $(x_{0}, y_{0})$
If $D > 0$ and $f_{x x} (x_{0}, y_{0}) < 0$ , then f has a local maximum at $(x_{0}, y_{0})$
If $D < 0$ , then f has a saddle point
If $D = 0$ , then the test fails.