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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|y| (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 x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ |x|y| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). = if (x, y) = (0,0),

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 52E
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Question Four (a) Assume that g(x, y) is a function of the form x² + y² g(x, y) = { 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 xy x²+y² if (x, y) (0,0), f(x, y) = 0 if (x, y) = (0,0) is not differentiable at point (0,0). (c) Let z(x, y) = x 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). 0 if (x, y) = (0,0),
Transcribed Image Text:Question Four (a) Assume that g(x, y) is a function of the form x² + y² g(x, y) = { 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 xy x²+y² if (x, y) (0,0), f(x, y) = 0 if (x, y) = (0,0) is not differentiable at point (0,0). (c) Let z(x, y) = x 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). 0 if (x, y) = (0,0),
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