Let fi : R" –→ R and f2 : R" → R be two functions. (5.1) Consider the following proposition: If epi(f1) N epi(f2) is a convex set, then at least one of the functions fi and f2 is a convex function. Either prove this proposition or give a counterexample (i.e., find two functions fi : R" → R and f2 : R" → R that satisfy the hypothesis but do not satisfy the conclusion).
Let fi : R" –→ R and f2 : R" → R be two functions. (5.1) Consider the following proposition: If epi(f1) N epi(f2) is a convex set, then at least one of the functions fi and f2 is a convex function. Either prove this proposition or give a counterexample (i.e., find two functions fi : R" → R and f2 : R" → R that satisfy the hypothesis but do not satisfy the conclusion).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 26E: 26. Let and. Prove that for any subset of T of .
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please give a counterexample
Transcribed Image Text:Let fi : R" –→ R and f2 : R" → R be two functions.
(5.1) Consider the following proposition:
If epi(f1) N epi(f2) is a convex set, then at least one of the functions f1 and f2 is a
convex function.
Either prove this proposition or give a counterexample (i.e., find two functions f1 : R" → R
and f2 : R" → R that satisfy the hypothesis but do not satisfy the conclusion).
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