1. Show that the following functions are convex by verifying the definition, i.e., that f(x + (1-X)y) ≤ f(x) + (1-x)f(y) is satisfied for all x, y in the domain of f and all A = [0, 1]: (a) f(u) = 1, u > 0, (b) f(u) = lu, u € R.
1. Show that the following functions are convex by verifying the definition, i.e., that f(x + (1-X)y) ≤ f(x) + (1-x)f(y) is satisfied for all x, y in the domain of f and all A = [0, 1]: (a) f(u) = 1, u > 0, (b) f(u) = lu, u € R.
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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![Certainly! Below is a transcription of the image for an educational website, along with an explanation of any mathematical notations:
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### Convex Function Verification
1. **Show that the following functions are convex by verifying the definition**, i.e., verify that
\[
f(\lambda x + (1 - \lambda)y) \leq \lambda f(x) + (1 - \lambda)f(y)
\]
is satisfied for all \( x, y \) in the domain of \( f \) and all \( \lambda \in [0, 1] \):
(a) \( f(u) = \frac{1}{u}, \, u > 0 \),
(b) \( f(u) = |u|, \, u \in \mathbb{R} \).
2. **Show that the following functions are convex by verifying the condition** that
\[
\nabla^2 f(x) \succeq 0
\]
is satisfied for all \( x \) in the domain of \( f \):
(a) \( f(u_1, u_2) = \ln(e^{u_1} + e^{u_2}) \),
(b) \( f(u_1, u_2, u_3, u_4) = \ln(1 - u_1 - u_2 - u_3 - u_4) \) over the domain \( \{u \in \mathbb{R}^4 | u_1 + u_2 + u_3 + u_4 \leq 1\} \).
3. **Use the definition of a convex set to show that if \( S_1 \) and \( S_2 \) are convex sets in \( \mathbb{R}^{m+n} \), then so is their partial sum**
\[
S = \{(x, y_1 + y_2) | x \in \mathbb{R}^m, y_1, y_2 \in \mathbb{R}^n; (x, y_1) \in S_1, (x, y_2) \in S_2\}.
\]
### Explanation of Mathematical Notations:
- The notation \( f(\lambda x + (1 - \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4589616-f4fd-4eb5-854c-a0440b9d838b%2F5991ef55-ebd4-4415-a50c-9263c5d3f4d0%2Fzeocvtl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Below is a transcription of the image for an educational website, along with an explanation of any mathematical notations:
---
### Convex Function Verification
1. **Show that the following functions are convex by verifying the definition**, i.e., verify that
\[
f(\lambda x + (1 - \lambda)y) \leq \lambda f(x) + (1 - \lambda)f(y)
\]
is satisfied for all \( x, y \) in the domain of \( f \) and all \( \lambda \in [0, 1] \):
(a) \( f(u) = \frac{1}{u}, \, u > 0 \),
(b) \( f(u) = |u|, \, u \in \mathbb{R} \).
2. **Show that the following functions are convex by verifying the condition** that
\[
\nabla^2 f(x) \succeq 0
\]
is satisfied for all \( x \) in the domain of \( f \):
(a) \( f(u_1, u_2) = \ln(e^{u_1} + e^{u_2}) \),
(b) \( f(u_1, u_2, u_3, u_4) = \ln(1 - u_1 - u_2 - u_3 - u_4) \) over the domain \( \{u \in \mathbb{R}^4 | u_1 + u_2 + u_3 + u_4 \leq 1\} \).
3. **Use the definition of a convex set to show that if \( S_1 \) and \( S_2 \) are convex sets in \( \mathbb{R}^{m+n} \), then so is their partial sum**
\[
S = \{(x, y_1 + y_2) | x \in \mathbb{R}^m, y_1, y_2 \in \mathbb{R}^n; (x, y_1) \in S_1, (x, y_2) \in S_2\}.
\]
### Explanation of Mathematical Notations:
- The notation \( f(\lambda x + (1 - \
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