(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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(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),](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ab93dff-ad88-4a81-9a2a-7598cfa6d89c%2Fdada7b43-2e8e-460d-a0d8-265eb085e50b%2Fjeysjh_processed.jpeg&w=3840&q=75)
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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