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 + ¹²), (b) f(u1, U2, U3, U4) = ln(1 -u₁ - 2 - u3 - u4) over the domain {u € R¹|u₁+U₂ + Uz + U₁ < 1}.
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 + ¹²), (b) f(u1, U2, U3, U4) = ln(1 -u₁ - 2 - u3 - u4) over the domain {u € R¹|u₁+U₂ + Uz + U₁ < 1}.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer question 2
![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) | xRm, 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%2F3dadd022-6f68-436a-98ca-cf787d439912%2Fsyccrn6_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) | xRm, y₁, y2 € R"; (x; y₁) € S₁, (x, y2) € S₂}.
Expert Solution
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