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 f1 : 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 f1 : R" → R and f2 : R" →R that satisfy the hypothesis but do not satisfy the conclusion).
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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it should have a counterexample, can you give?
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 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).
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