RR be a differentiable function. For any point (z,y) € R², we denote [(I,y)] B The Lagrange's theorem says that for any point A(z, y) and B(a, b) in R² we have Exercises f(z,y) = f(a,b) = f(A) – ƒ(B) = Jƒ(zª, y")[A – B] Of(x,y) 81 -(2-a) + Of(ty) By (y-b) where (z,y") is some point in the line segment AB. Given the inequality (aX+BY)²(a²+B²) (X² +Y2) for any a, 8, X, Y E R. Prove that (8f(r*. 2 f(x,y)-f(a,b)| ≤ [(**))² + (ºf (*^, '")) ². Consequently, if (°r,”)). V—a) + (y = b. Oy 2 (of (2.9))² + (0f (1,1))² then f is Lipschitz. M,V(I,y) ER²
RR be a differentiable function. For any point (z,y) € R², we denote [(I,y)] B The Lagrange's theorem says that for any point A(z, y) and B(a, b) in R² we have Exercises f(z,y) = f(a,b) = f(A) – ƒ(B) = Jƒ(zª, y")[A – B] Of(x,y) 81 -(2-a) + Of(ty) By (y-b) where (z,y") is some point in the line segment AB. Given the inequality (aX+BY)²(a²+B²) (X² +Y2) for any a, 8, X, Y E R. Prove that (8f(r*. 2 f(x,y)-f(a,b)| ≤ [(**))² + (ºf (*^, '")) ². Consequently, if (°r,”)). V—a) + (y = b. Oy 2 (of (2.9))² + (0f (1,1))² then f is Lipschitz. M,V(I,y) ER²
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
![4 Let R²R be a differentiable function. For any point (7, y) € R², we denote
1:1
4
The Lagrange's theorem says that for any point A(z, y) and B(a, b) in R² we have
f(z,y) — f(a,b) = f(A) — ƒ(B) = Jƒ(z,y')[A – B]
of (x¹, y) (y - b)
Oy
Exercises
[(t, y)]=
Of(x,y)
81
Consequently, if
-(1-a) +
where (z,y) is some point in the line segment AB.
Given the inequality (aX+BY)²(a² + B²) (X²+Y²) for any a, 8, X, Y E R. Prove
that
2
f(x,y)-f(a,b)| ≤₁ (°F (**, *))² + (
of(x,y)
Oy
2
2
(of(x, y))² + (0f (1,1))²
2
EEN
<M, V(I, y) € R²
a)² + (y-b)².
then f is Lipschitz.
5. Using the above inequality, and note that for any point A(z, y), B(a, b) e R², we have
||AB||||(r. y) - (a,b)
√(ra)² + (y-b)²,
prove that ln(1 + y²), sinr cos y, e
are Lipschitz functions.
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0aad5e5-5f72-4448-ac36-adf134944ef5%2Fe7cd0585-2e63-492e-9938-a3cfa850842d%2F1oa3ve8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4 Let R²R be a differentiable function. For any point (7, y) € R², we denote
1:1
4
The Lagrange's theorem says that for any point A(z, y) and B(a, b) in R² we have
f(z,y) — f(a,b) = f(A) — ƒ(B) = Jƒ(z,y')[A – B]
of (x¹, y) (y - b)
Oy
Exercises
[(t, y)]=
Of(x,y)
81
Consequently, if
-(1-a) +
where (z,y) is some point in the line segment AB.
Given the inequality (aX+BY)²(a² + B²) (X²+Y²) for any a, 8, X, Y E R. Prove
that
2
f(x,y)-f(a,b)| ≤₁ (°F (**, *))² + (
of(x,y)
Oy
2
2
(of(x, y))² + (0f (1,1))²
2
EEN
<M, V(I, y) € R²
a)² + (y-b)².
then f is Lipschitz.
5. Using the above inequality, and note that for any point A(z, y), B(a, b) e R², we have
||AB||||(r. y) - (a,b)
√(ra)² + (y-b)²,
prove that ln(1 + y²), sinr cos y, e
are Lipschitz functions.
2
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