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Let f: R² → R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).

Let f: R² → R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
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Real Analysis II Please find idea for problem from sample in 2nd photo
10:41 ← Expert Answer Ⓒ Step1 a) ⇒ Step2 b) (a) af dx (c) af ay =) = (b) f(c) + L ((x, y) = c) and Hence, f (x, y) = = => z = D f(c) = (cat)) ale) 2x g= 2 y əg ax =) 93 az 8+ (4,4) (x-2, 4-2) 8 + 4(x-2) + 4 (5-2). 2 x² + y² -1 equation of = a ) x² + y2 x² + y²_z = 0 a = 4 af ax at 10 = 4x + 4y-2 = 8 ? = (4,4) (2² +2²)+L((x,y)-(2,23) 8 + L ((x-2, y-2)) 23 25 = 23 ay C = (2,2) = 4 f(c) = 8 tangent plane 4 (x-2) + + (y-2) -1 (2-8) = 0 ag => - 3x1.- ax IS 29 dy √x = 4 Equation of this plane (x-) +33 (5-2) -1 (2-0) = 0 8
Transcribed Image Text:10:41 ← Expert Answer Ⓒ Step1 a) ⇒ Step2 b) (a) af dx (c) af ay =) = (b) f(c) + L ((x, y) = c) and Hence, f (x, y) = = => z = D f(c) = (cat)) ale) 2x g= 2 y əg ax =) 93 az 8+ (4,4) (x-2, 4-2) 8 + 4(x-2) + 4 (5-2). 2 x² + y² -1 equation of = a ) x² + y2 x² + y²_z = 0 a = 4 af ax at 10 = 4x + 4y-2 = 8 ? = (4,4) (2² +2²)+L((x,y)-(2,23) 8 + L ((x-2, y-2)) 23 25 = 23 ay C = (2,2) = 4 f(c) = 8 tangent plane 4 (x-2) + + (y-2) -1 (2-8) = 0 ag => - 3x1.- ax IS 29 dy √x = 4 Equation of this plane (x-) +33 (5-2) -1 (2-0) = 0 8
3. Let f: R2R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).
Transcribed Image Text:3. Let f: R2R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove that Df(0, 0) exists. Note: Df(c) only exists at c = (0,0) and f is continuous only at (0,0).
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