(a) (only 4115 students) Prove that the function ƒ (x₁, x₂) = log (e™¹ + e²²) is convex on R².
(a) (only 4115 students) Prove that the function ƒ (x₁, x₂) = log (e™¹ + e²²) is convex on R².
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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![### Problem Statement
**1. (a) (only 4115 students)** Prove that the function
\[ f(x_1, x_2) = \log(e^{x_1} + e^{x_2}) \]
is convex on \(\mathbb{R}^2\).
### Explanation:
This problem asks students to prove the convexity of a specific function, \( f(x_1, x_2) \), on the two-dimensional real number space, denoted as \(\mathbb{R}^2\). The function is defined as the natural logarithm of the sum of two exponential functions, \( e^{x_1} \) and \( e^{x_2} \).
### Understanding Convexity:
A function \( f \) is convex on a set if, for any two points \( \mathbf{x}, \mathbf{y} \) in that set and for any \( \lambda \) satisfying \( 0 \leq \lambda \leq 1 \), the following inequality holds:
\[ f(\lambda \mathbf{x} + (1-\lambda) \mathbf{y}) \leq \lambda f(\mathbf{x}) + (1-\lambda) f(\mathbf{y}) \]
To prove convexity, one can utilize the second-order conditions, which involve ensuring that the Hessian matrix of the function is positive semi-definite at all points in the domain.
No graphs or diagrams are included in this problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde27b5cb-cd0c-4d35-bbc9-1c5df406bf15%2F0878dff5-2788-4491-b4b6-242cbfa0ba63%2F4sh486_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**1. (a) (only 4115 students)** Prove that the function
\[ f(x_1, x_2) = \log(e^{x_1} + e^{x_2}) \]
is convex on \(\mathbb{R}^2\).
### Explanation:
This problem asks students to prove the convexity of a specific function, \( f(x_1, x_2) \), on the two-dimensional real number space, denoted as \(\mathbb{R}^2\). The function is defined as the natural logarithm of the sum of two exponential functions, \( e^{x_1} \) and \( e^{x_2} \).
### Understanding Convexity:
A function \( f \) is convex on a set if, for any two points \( \mathbf{x}, \mathbf{y} \) in that set and for any \( \lambda \) satisfying \( 0 \leq \lambda \leq 1 \), the following inequality holds:
\[ f(\lambda \mathbf{x} + (1-\lambda) \mathbf{y}) \leq \lambda f(\mathbf{x}) + (1-\lambda) f(\mathbf{y}) \]
To prove convexity, one can utilize the second-order conditions, which involve ensuring that the Hessian matrix of the function is positive semi-definite at all points in the domain.
No graphs or diagrams are included in this problem.
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