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(a) Assume that g(x, y) is a function of the form 2+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ x|ly| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). if (x, y) = (0,0),

(a) Assume that g(x, y) is a function of the form 2+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ x|ly| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). if (x, y) = (0,0),

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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(a) Assume that g(x, y) is a function of the form x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ x|ly| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). (b) Show that the function defined by ay if (x, y) = (0,0), x²+y² f(x, y) = {} 0 if (x, y) = (0,0) is not differentiable at point (0,0). (c) Let z(x, y) = xy be a function defined on a disk D in the positive quadrant containing the point (1,2). Prove whether z(x, y) satisfies the Clairaut Theorem at (1,2). if (x, y) = (0,0),
Transcribed Image Text:(a) Assume that g(x, y) is a function of the form x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ x|ly| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). (b) Show that the function defined by ay if (x, y) = (0,0), x²+y² f(x, y) = {} 0 if (x, y) = (0,0) is not differentiable at point (0,0). (c) Let z(x, y) = xy be a function defined on a disk D in the positive quadrant containing the point (1,2). Prove whether z(x, y) satisfies the Clairaut Theorem at (1,2). if (x, y) = (0,0),
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