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(a) Prove that, if f,g: R → R are convex functions and t≥ 0, then the function h(x) f(x) + tg(x) is convex. =

(a) Prove that, if f,g: R → R are convex functions and t≥ 0, then the function h(x) f(x) + tg(x) is convex. =

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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Mathematics - Optimization 

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(a) Prove that, if f,g: R → R are convex functions and t≥ 0, then the function h(x) f(x) + tg(x) is convex. = (b) For which triples (a, b, c) of real numbers is the function f(x) = ax² + bx+c convex? Prove your answer without using any calculus. (c) For which integers n ≥ 1 is the function f(x) = = xn convex? You may use the fact that a function f: R→ R such that f"(x) ≥ 0 for all x is convex, but not the converse of this fact. (d) Is it true that the product of two convex functions f, g: R → R is always convex? Prove or give a counterexample. (e) Bonus: Give an example of a function f : R → R such that ƒ(1x1 + x₂) ≤ ½⁄ƒ(x1) + ½ ƒ (x2) for all x1, x2 € R, but f is not convex. (Hint: this is a very very weird function. The fact that R is a vector space over Q is relevant to its construction.).
Transcribed Image Text:(a) Prove that, if f,g: R → R are convex functions and t≥ 0, then the function h(x) f(x) + tg(x) is convex. = (b) For which triples (a, b, c) of real numbers is the function f(x) = ax² + bx+c convex? Prove your answer without using any calculus. (c) For which integers n ≥ 1 is the function f(x) = = xn convex? You may use the fact that a function f: R→ R such that f"(x) ≥ 0 for all x is convex, but not the converse of this fact. (d) Is it true that the product of two convex functions f, g: R → R is always convex? Prove or give a counterexample. (e) Bonus: Give an example of a function f : R → R such that ƒ(1x1 + x₂) ≤ ½⁄ƒ(x1) + ½ ƒ (x2) for all x1, x2 € R, but f is not convex. (Hint: this is a very very weird function. The fact that R is a vector space over Q is relevant to its construction.).
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