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7. Consider the function f(x) defined on the set by D = {x = R3: x > 0, Xz > 0, Xg >0} = f(x) = (x₁)¹ + (x₂)²² + (x3)²³, where ri> 0 for i 1, 2, 3. = 1, 2, 3. (a) Show that f(x) is strictly convex on D if r; > 1 for i (b) Show that f(x) is strictly concave on D if 0 < r; < 1 for i = 1, 2, 3.

7. Consider the function f(x) defined on the set by D = {x = R3: x > 0, Xz > 0, Xg >0} = f(x) = (x₁)¹ + (x₂)²² + (x3)²³, where ri> 0 for i 1, 2, 3. = 1, 2, 3. (a) Show that f(x) is strictly convex on D if r; > 1 for i (b) Show that f(x) is strictly concave on D if 0 < r; < 1 for i = 1, 2, 3.

Advanced Engineering Mathematics
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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7. Consider the function f(x) defined on the set by D = {x = R3: x, > 0, Xz > 0, Xg >0} x3 f(x) = (x₁)²¹ + (X₂)′² + (x3)′³, where r> 0 for i = 1, 2, 3. = 1, 2, 3. (a) Show that f(x) is strictly convex on D if r; > 1 for i (b) Show that f(x) is strictly concave on D if 0 < r; < 1 for i = = 1, 2, 3.
Transcribed Image Text:7. Consider the function f(x) defined on the set by D = {x = R3: x, > 0, Xz > 0, Xg >0} x3 f(x) = (x₁)²¹ + (X₂)′² + (x3)′³, where r> 0 for i = 1, 2, 3. = 1, 2, 3. (a) Show that f(x) is strictly convex on D if r; > 1 for i (b) Show that f(x) is strictly concave on D if 0 < r; < 1 for i = = 1, 2, 3.
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