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For each of the following functions determine whether it is convex, concave, quasiconvex, or quasiconcave. (a) f(x)= e²-1 on R. (b) f(x1, x2) = x122 on R²+. (c) f(x1, x2) = 1/(x122) on R²+. (d) f(x1,₂)= 1/2 on R²+.

For each of the following functions determine whether it is convex, concave, quasiconvex, or quasiconcave. (a) f(x)= e²-1 on R. (b) f(x1, x2) = x122 on R²+. (c) f(x1, x2) = 1/(x122) on R²+. (d) f(x1,₂)= 1/2 on R²+.

Advanced Engineering Mathematics
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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I need help with part d. I only need to know if it is convex or concave. 

**Title: Analyzing Convexity and Concavity of Functions** For each of the following functions, determine whether it is convex, concave, quasiconvex, or quasiconcave. (a) \( f(x) = e^x - 1 \) on \( \mathbb{R} \). (b) \( f(x_1, x_2) = x_1 x_2 \) on \( \mathbb{R}^2_{++} \). (c) \( f(x_1, x_2) = 1/(x_1 x_2) \) on \( \mathbb{R}^2_{++} \). (d) \( f(x_1, x_2) = x_1/x_2 \) on \( \mathbb{R}^2_{++} \). **Explanation**: - **Convex Function**: A function is convex if its domain is a convex set and if for any two points within the domain, the function evaluated at the weighted average of these points is less than or equal to the weighted average of the function evaluated at these points. - **Concave Function**: A function is concave if its domain is a convex set and if for any two points within the domain, the function evaluated at the weighted average of these points is greater than or equal to the weighted average of the function evaluated at these points. - **Quasiconvex Function**: A function is quasiconvex if all its sublevel sets are convex sets. - **Quasiconcave Function**: A function is quasiconcave if all its superlevel sets are convex sets. **Variables**: - \( \mathbb{R} \) denotes the set of all real numbers. - \( \mathbb{R}^2_{++} \) denotes the set of all pairs of positive real numbers.
Transcribed Image Text:**Title: Analyzing Convexity and Concavity of Functions** For each of the following functions, determine whether it is convex, concave, quasiconvex, or quasiconcave. (a) \( f(x) = e^x - 1 \) on \( \mathbb{R} \). (b) \( f(x_1, x_2) = x_1 x_2 \) on \( \mathbb{R}^2_{++} \). (c) \( f(x_1, x_2) = 1/(x_1 x_2) \) on \( \mathbb{R}^2_{++} \). (d) \( f(x_1, x_2) = x_1/x_2 \) on \( \mathbb{R}^2_{++} \). **Explanation**: - **Convex Function**: A function is convex if its domain is a convex set and if for any two points within the domain, the function evaluated at the weighted average of these points is less than or equal to the weighted average of the function evaluated at these points. - **Concave Function**: A function is concave if its domain is a convex set and if for any two points within the domain, the function evaluated at the weighted average of these points is greater than or equal to the weighted average of the function evaluated at these points. - **Quasiconvex Function**: A function is quasiconvex if all its sublevel sets are convex sets. - **Quasiconcave Function**: A function is quasiconcave if all its superlevel sets are convex sets. **Variables**: - \( \mathbb{R} \) denotes the set of all real numbers. - \( \mathbb{R}^2_{++} \) denotes the set of all pairs of positive real numbers.
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