4. Prove that a subset of R is convex if and only if it is an interval. 5. Show that the union of two convex sets is not a convex set. 7. Given v ∈ R3 0 and ϵ > 0 then let K(v, ϵ) = {x ∈ R3 : ϵ∥v∥∥x∥ ≤ ⟨v, x⟩}. show that K(v, ϵ) is convex. 13. Consider f a convex function and S a convex set. Try that X∗ = {x∗ : f(x∗) ≤ f(x), ∀x ∈ S} It is convex. Please give the answers step by step as explicit as possible. thank you

4. Prove that a subset of R is convex if and only if it is an interval. 5. Show that the union of two convex sets is not a convex set. 7. Given v ∈ R3 0 and ϵ > 0 then let K(v, ϵ) = {x ∈ R3 : ϵ∥v∥∥x∥ ≤ ⟨v, x⟩}. show that K(v, ϵ) is convex. 13. Consider f a convex function and S a convex set. Try that X∗ = {x∗ : f(x∗) ≤ f(x), ∀x ∈ S} It is convex. Please give the answers step by step as explicit as possible. thank you

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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