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xy - 1. Let f: R² → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = ² + y² if (x, y) = (0,0). - (a) Prove that Duf(0, 0) = 0 if u = (1,0). (b) Prove that Duf(0,0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0.

xy - 1. Let f: R² → R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = ² + y² if (x, y) = (0,0). - (a) Prove that Duf(0, 0) = 0 if u = (1,0). (b) Prove that Duf(0,0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0.

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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Real Analysis II Solve using full derivative instead of partial
xy 1. Let f: R²R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = x² + y² if (x, y) = (0,0). (a) Prove that Duf(0,0) = 0 if u = (1,0). - (b) Prove that Duf(0,0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0. (d) Prove that Df(0, 0) does not exist (this can be done by proving that f is not continuous at (0,0)).
Transcribed Image Text:xy 1. Let f: R²R where f(x, y) = 0 if (x, y) = (0,0) and f(x, y) = x² + y² if (x, y) = (0,0). (a) Prove that Duf(0,0) = 0 if u = (1,0). - (b) Prove that Duf(0,0) = 0 if u = (0, 1). (c) Prove that Duf(0, 0) does not exist if u = (a, b) where ab 0. (d) Prove that Df(0, 0) does not exist (this can be done by proving that f is not continuous at (0,0)).
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