Given a function f(x) = f(x1, x2) = (x2 − x 2 1 )(x2 − 2x 2 1 ), where x ∈ S = [x1, x2] T : x1 ∈ R, x2 < 0 . (a) Is the set S open or closed or both or neither? Is it bounded? Is it convex or concave or neither? (Note: No steps are required.) (b) Determine the gradient vector ∇f(x) and the Hessian matrix ∇2f(x) of f(x) over S. (c) Prove that f(x) is a strictly convex function over the set S.
Given a function f(x) = f(x1, x2) = (x2 − x 2 1 )(x2 − 2x 2 1 ), where x ∈ S = [x1, x2] T : x1 ∈ R, x2 < 0 . (a) Is the set S open or closed or both or neither? Is it bounded? Is it convex or concave or neither? (Note: No steps are required.) (b) Determine the gradient vector ∇f(x) and the Hessian matrix ∇2f(x) of f(x) over S. (c) Prove that f(x) is a strictly convex function over the set S.
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
Given a function f(x) = f(x1, x2) = (x2 − x
2
1
)(x2 − 2x
2
1
), where x ∈ S =
[x1, x2]
T
:
x1 ∈ R, x2 < 0
.
(a) Is the set S open or closed or both or neither? Is it bounded? Is it convex or concave or neither? (Note: No steps are required.)
(b) Determine the gradient
(c) Prove that f(x) is a strictly convex function over the set S.
![Question 4 (50%)
Given a function f(x) = f(x1, x2)
X1 € R, x2 < 0}.
(a) Is the set S open or closed or both or neither? Is it bounded? Is it convex or concave
or neither? (Note: No steps are required.)
= (x2 – xỉ)(x2 – 2x}), where x € S =
{{#1,#2]" :
-
(b) Determine the gradient vector Vf (x) and the Hessian matrix V²f(x) of f (x) over S.
(c) Prove that f (x) is a strictly convex function over the set S.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb22d8e54-f2b3-424b-b0d7-ac33bf8b963f%2Fbce91e46-f89d-4615-aad1-c551b49d1602%2Fiz8qb3_processed.png&w=3840&q=75)
Transcribed Image Text:Question 4 (50%)
Given a function f(x) = f(x1, x2)
X1 € R, x2 < 0}.
(a) Is the set S open or closed or both or neither? Is it bounded? Is it convex or concave
or neither? (Note: No steps are required.)
= (x2 – xỉ)(x2 – 2x}), where x € S =
{{#1,#2]" :
-
(b) Determine the gradient vector Vf (x) and the Hessian matrix V²f(x) of f (x) over S.
(c) Prove that f (x) is a strictly convex function over the set S.
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