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Let f(x) be a convex function on R¹. Let a be a fixed vector on R" and let x be a fixed real number. Define g(x) on R" by g(x) = f(a x + x). Show g(x) is convex on R" but that if n ≥ 2, then g(x) is not strictly convex. Deduce that g(x, y, z) = (4x + 5y - 8z + 17)8 is convex but not strictly convex on R³. State why this does not follow from Theorem (2.3.10)(c).

Let f(x) be a convex function on R¹. Let a be a fixed vector on R" and let x be a fixed real number. Define g(x) on R" by g(x) = f(a x + x). Show g(x) is convex on R" but that if n ≥ 2, then g(x) is not strictly convex. Deduce that g(x, y, z) = (4x + 5y - 8z + 17)8 is convex but not strictly convex on R³. State why this does not follow from Theorem (2.3.10)(c).

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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24. Let f(x) be a convex function on R¹. Let a be a fixed vector on R" and let x be a fixed real number. Define g(x) on R" by g(x) = f(a x + x). Show g(x) is convex on R" but that if n ≥ 2, then g(x) is not strictly convex. Deduce that g(x, y, z) = (4x + 5y - 8z + 17)8 is convex but not strictly convex on R³. State why this does not follow from Theorem (2.3.10) (c).
Transcribed Image Text:24. Let f(x) be a convex function on R¹. Let a be a fixed vector on R" and let x be a fixed real number. Define g(x) on R" by g(x) = f(a x + x). Show g(x) is convex on R" but that if n ≥ 2, then g(x) is not strictly convex. Deduce that g(x, y, z) = (4x + 5y - 8z + 17)8 is convex but not strictly convex on R³. State why this does not follow from Theorem (2.3.10) (c).
(2.3.10) Theorem. (a) Iƒ ƒ₁(x), ..., f₁(x) are convex functions on a convex set C in R", then f(x) = f₁(x) + ƒ₂(x) + + fx(x) is convex. Moreover, if at least one f(x) is strictly convex on C, then the sum f(x) is strictly convex. (b) If f(x) is convex (resp. strictly convex) on a convex set C in R" and if x is a positive number, then af (x) is convex (resp. strictly convex) on C. (c) If f(x) is a convex (resp. strictly convex) function defined on a convex set C in R", and if g(y) is an increasing (resp. strictly increasing) convex function defined on the range of f(x) in R, then the composite function g(f(x)) is convex (resp. strictly convex) on C.
Transcribed Image Text:(2.3.10) Theorem. (a) Iƒ ƒ₁(x), ..., f₁(x) are convex functions on a convex set C in R", then f(x) = f₁(x) + ƒ₂(x) + + fx(x) is convex. Moreover, if at least one f(x) is strictly convex on C, then the sum f(x) is strictly convex. (b) If f(x) is convex (resp. strictly convex) on a convex set C in R" and if x is a positive number, then af (x) is convex (resp. strictly convex) on C. (c) If f(x) is a convex (resp. strictly convex) function defined on a convex set C in R", and if g(y) is an increasing (resp. strictly increasing) convex function defined on the range of f(x) in R, then the composite function g(f(x)) is convex (resp. strictly convex) on C.
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