If f(x) is Lipschitz with constant L and |f(x)| < B, then [f(x)]² is Lipschitz with constant 2LB.
If f(x) is Lipschitz with constant L and |f(x)| < B, then [f(x)]² is Lipschitz with constant 2LB.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![1. Determine if the following statements are TRUE or FALSE, and justify your answer
(e.g. a short proof or counter-example, refer to a theorem, etc).
a) If f(x) is Lipschitz with constant L and |f(x)| < B, then [f(x)]² is Lipschitz with
constant 2LB.
b) Consider data points (x;, Yi) and assume they fit the model y(x)
parameter a can then be estimated by the slope in a log-log plot of the points.
c) Consider two n-by-n matrices A, U, where A is general but U is upper-triangular
(with positive diagonal entries). The total operation count of computing the
matrix-vector product Ax is lower than solving the linear system Ux
backward substitution, for some vectors x,b.
b. a". The
b using
a b 0
d) The matrix A:
ba b is positive definite if and only if a > 0 and |b| < a/V2.
0 b a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F185c69b8-ae2a-4395-806d-86dda15e2466%2Fe6e37662-e8ba-44a9-ac36-0e590f0f1c89%2Fwfssh_processed.png&w=3840&q=75)
Transcribed Image Text:1. Determine if the following statements are TRUE or FALSE, and justify your answer
(e.g. a short proof or counter-example, refer to a theorem, etc).
a) If f(x) is Lipschitz with constant L and |f(x)| < B, then [f(x)]² is Lipschitz with
constant 2LB.
b) Consider data points (x;, Yi) and assume they fit the model y(x)
parameter a can then be estimated by the slope in a log-log plot of the points.
c) Consider two n-by-n matrices A, U, where A is general but U is upper-triangular
(with positive diagonal entries). The total operation count of computing the
matrix-vector product Ax is lower than solving the linear system Ux
backward substitution, for some vectors x,b.
b. a". The
b using
a b 0
d) The matrix A:
ba b is positive definite if and only if a > 0 and |b| < a/V2.
0 b a
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