Let A C R" be a convex set. In Week 4, we have learned the following definition for quasi-concave functions: Definition 1. f : A → R is a quasi-concave function if and only if for any c E R, the set {x € A| f(x) > c} is convex. Here is an alternative definition: Definition 2. f : A →R is a quasi-concave function if and only if for any x1, x2 E A and any a € (0, 1), f (ax1 +(1– a)x2) > min {f (x1),f (x2)}. Please prove that these two definitions are equivalent. Hint: To show equivalence, you need to prove two parts: (1) if ƒ satisfies the condition in Definition 1, then it must satisfy the condition in Definition 2; (2) if ƒ satisfies the condition in Definition 2, then it must satisfy the condition in Definition 1.
Let A C R" be a convex set. In Week 4, we have learned the following definition for quasi-concave functions: Definition 1. f : A → R is a quasi-concave function if and only if for any c E R, the set {x € A| f(x) > c} is convex. Here is an alternative definition: Definition 2. f : A →R is a quasi-concave function if and only if for any x1, x2 E A and any a € (0, 1), f (ax1 +(1– a)x2) > min {f (x1),f (x2)}. Please prove that these two definitions are equivalent. Hint: To show equivalence, you need to prove two parts: (1) if ƒ satisfies the condition in Definition 1, then it must satisfy the condition in Definition 2; (2) if ƒ satisfies the condition in Definition 2, then it must satisfy the condition in Definition 1.
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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1
Transcribed Image Text:Let A CR" be a convex set. In Week 4, we have learned the following definition
for quasi-concave functions:
Definition 1. f : A → R is a quasi-concave function if and only if for any c E R, the set
{x € A| f(x) > c} is convex.
Here is an alternative definition:
Definition 2. f: A
→ R is a quasi-concave function if and only if for any x1, x2 E A and any
a E (0, 1), ƒ (ax1 +(1– a)x2) > min {f (x1), ƒ (x2)}.
Please prove that these two definitions are equivalent.
Hint: To show equivalence, you need to prove two parts: (1) if f satisfies the condition in
Definition 1, then it must satisfy the condition in Definition 2; (2) if ƒ satisfies the condition
in Definition 2, then it must satisfy the condition in Definition 1.
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