(a) Assume that g(x, y) is a function of the formg(x, y){0 if (x, y) = (0,0).(i) Show thatg(x, y)| ≤|xl|y|(ii) Using part (i), prove that g(x, y) is continuous at point (0, 0).(b) Show that the function defined byxy√²+² 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 quadrantcontaining the point (1,2). Prove whether z(x, y) satisfies the Clairaut Theoremat (1, 2).=if (x, y) = (0,0),

Answered: Image /qna-images/answer/129d0a16-4866-40a0-bb7a-33efbd4633c1.jpg

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

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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Question

(a) Assume that g(x, y) is a function of the form
g(x, y)
{
0 if (x, y) = (0,0).
(i) Show that
g(x, y)| ≤|xl|y|
(ii) Using part (i), prove that g(x, y) is continuous at point (0, 0).
(b) Show that the function defined by
xy
√²+² 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).
=
if (x, y) = (0,0),

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