1. Show that the following functions are convex by verifying the definition, i.e., thatf(Xr+ (1-X)y) ≤ f(x) + (1-X)f(y)is satisfied for all r, y in the domain of f and all λ = [0, 1]:(a) f(u) = 1, u > 0,(b) f(u) = \u, u R.2. Show that the following functions are convex by verifying the condition that² f(x) > 0is satisfied for all a in the domain of f:(a) f(u₁, u₂) In(e" + e"),=(b) f(u₁, U2, U3, U4) = ln(1-u₁-uz-us-u4) over the domain {u E R4|u₁ + ₂ + us+ us ≤ 1}.3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in R+", then so is theirpartial sumS = {(x,y₁ + y2) | ER", ₁,92 € R"; (a; y₁) S₁, (x, y2) € S₂}.

Since you have posted multiple questions, we will provide the solution only to the first question as…

Show that the following functions are convex by verifying the definition, i.e., that. f(x + (1-A)y) ≤ f(x) + (1 - A)f(y) is satisfied for all x, y in the domain of f and all A = [0, 1]: (a) f(u) = 1,u> 0, (b) f(u) u, u € R. =

Show that the following functions are convex by verifying the definition, i.e., that. f(x + (1-A)y) ≤ f(x) + (1 - A)f(y) is satisfied for all x, y in the domain of f and all A = [0, 1]: (a) f(u) = 1,u> 0, (b) f(u) u, u € R. =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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Question

1. Show that the following functions are convex by verifying the definition, i.e., that
f(Xr+ (1-X)y) ≤ f(x) + (1-X)f(y)
is satisfied for all r, y in the domain of f and all λ = [0, 1]:
(a) f(u) = 1, u > 0,
(b) f(u) = \u, u R.
2. Show that the following functions are convex by verifying the condition that
² f(x) > 0
is satisfied for all a in the domain of f:
(a) f(u₁, u₂) In(e" + e"),
=
(b) f(u₁, U2, U3, U4) = ln(1-u₁-uz-us-u4) over the domain {u E R4|u₁ + ₂ + us+ us ≤ 1}.
3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in R+", then so is their
partial sum
S = {(x,y₁ + y2) | ER", ₁,92 € R"; (a; y₁) S₁, (x, y2) € S₂}.

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