Question 5. a) Let fi, i E I be a collection of convex functions defined on a convex set 2. Show that the function f defined as f(x):= supiel fi(x) is convex on the region where it is finite. b) Use the above result to show that, for s = [0, 1], the function f(x) ᎦᏆ . x for a ≤0 for r> 0 is convex. Is f strictly convex? c) Use the above result also to show that the function f : R" → R defined by |||2 ||T|| when |||| ≥ 1 when |||| < 1 f(x) = is convex, where || || is the standard Euclidean norm in R". Hint: Show first that norms are convex and then write f as maximum of two convex functions. Is f strictly convex?
Question 5. a) Let fi, i E I be a collection of convex functions defined on a convex set 2. Show that the function f defined as f(x):= supiel fi(x) is convex on the region where it is finite. b) Use the above result to show that, for s = [0, 1], the function f(x) ᎦᏆ . x for a ≤0 for r> 0 is convex. Is f strictly convex? c) Use the above result also to show that the function f : R" → R defined by |||2 ||T|| when |||| ≥ 1 when |||| < 1 f(x) = is convex, where || || is the standard Euclidean norm in R". Hint: Show first that norms are convex and then write f as maximum of two convex functions. Is f strictly convex?
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 5. a) Let fi, i e I be a collection of convex functions defined
on a convex set N. Show that the function f defined as f(x) := sup;eI fi(x) is
convex on the region where it is finite.
b) Use the above result to show that, for s e [0, 1], the function
for a <0
f(x) =
%3D
for x>0
is convex. Is f strictly convex?
c) Use the above result also to show that the function f : R" - R defined
by
Sa|? when ||æ|| 2 1
f(x) =
when ||æ|| < 1
is convex, where || · || is the standard Euclidean norm in R". Hint: Show first
that norms are convex and then write f as maximum of two convex functions.
Is f strictly convex?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8064b640-6afe-4ebd-b93a-f58737d12a2b%2F02e9b219-ac7d-4837-8e53-3e397ef8c6d5%2F2iwcv8r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 5. a) Let fi, i e I be a collection of convex functions defined
on a convex set N. Show that the function f defined as f(x) := sup;eI fi(x) is
convex on the region where it is finite.
b) Use the above result to show that, for s e [0, 1], the function
for a <0
f(x) =
%3D
for x>0
is convex. Is f strictly convex?
c) Use the above result also to show that the function f : R" - R defined
by
Sa|? when ||æ|| 2 1
f(x) =
when ||æ|| < 1
is convex, where || · || is the standard Euclidean norm in R". Hint: Show first
that norms are convex and then write f as maximum of two convex functions.
Is f strictly convex?
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