1. 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 X € [0, 1]: (a) f(u) = , u > 0, (b) f(u) = lu, u € R. 2. Show that the following functions are convex by verifying the condition that V² f(x) ≥ 0 is satisfied for all x in the domain of f: (a) f(u₁, U₂) = ln(e¹¹ + e¹¹2), (b) f(u1, U2, U3, U4) = n(1 - U₁ - ₂ - u3 - u4) over the domain {u € R¹ | U₁ + U₂ + Uz + U₁ < 1}. 3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in Rm+n, then so is their partial sum S = {(x, y₁ + y2) |xERm, y₁, y2 € R"; (x; y₁) E S₁, (x, y2) € S₂}.
1. 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 X € [0, 1]: (a) f(u) = , u > 0, (b) f(u) = lu, u € R. 2. Show that the following functions are convex by verifying the condition that V² f(x) ≥ 0 is satisfied for all x in the domain of f: (a) f(u₁, U₂) = ln(e¹¹ + e¹¹2), (b) f(u1, U2, U3, U4) = n(1 - U₁ - ₂ - u3 - u4) over the domain {u € R¹ | U₁ + U₂ + Uz + U₁ < 1}. 3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in Rm+n, then so is their partial sum S = {(x, y₁ + y2) |xERm, y₁, y2 € R"; (x; y₁) E S₁, (x, y2) € S₂}.
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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Please answer question 3
![1. 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 X € [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
V² f(x) ≥ 0
is satisfied for all x in the domain of f:
(a) f(u₁, u₂) = ln(e¹¹ + e¹²),
(b) f(u1, U2, U3, U4) = ln(1 − U₁ — U₂ - uz-u4) over the domain {u € Rª | u₁ + U2 + Uz + U₁ < 1}.
3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in Rm+n, then so is their
partial sum
S = {(x, y₁ + y2) | x Rm, y₁, y2 € R"; (x; y₁) € S₁, (x, y2) € S₂}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec417cb7-f4c3-4376-ad3a-c1f5cc0351ef%2Fda6b5784-32b4-49a5-8db8-6500cd33aadf%2Ft2vob1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. 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 X € [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
V² f(x) ≥ 0
is satisfied for all x in the domain of f:
(a) f(u₁, u₂) = ln(e¹¹ + e¹²),
(b) f(u1, U2, U3, U4) = ln(1 − U₁ — U₂ - uz-u4) over the domain {u € Rª | u₁ + U2 + Uz + U₁ < 1}.
3. Use the definition of a convex set to show that if S₁ and S₂ are convex sets in Rm+n, then so is their
partial sum
S = {(x, y₁ + y2) | x Rm, y₁, y2 € R"; (x; y₁) € S₁, (x, y2) € S₂}.
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