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[3.8] Let fi, f2,., fk: R" → R be convex functions. Consider the function f defined by f(x) =E=a;f;(x), where a; > 0 for j = 1, 2,.., k. Show that f is convex. State and prove a similar result for concave functions.

[3.8] Let fi, f2,., fk: R" → R be convex functions. Consider the function f defined by f(x) =E=a;f;(x), where a; > 0 for j = 1, 2,.., k. Show that f is convex. State and prove a similar result for concave functions.

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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[3.8] Let fi, f2,.., fk: R" → R be convex functions. Consider the function f defined by f(x)= E-a;f;(x), where a ; > 0 for j = 1, 2,..., k. Show that f is convex. State and prove a similar result for concave functions. [3.9] Let fi, f2,., fk: R" → R be convex functions. Consider the function f defined by f(x) = max{fj (x), fz(x),..., fx (x)}. Show that f is convex. State and prove a similar result for concave functions.
Transcribed Image Text:[3.8] Let fi, f2,.., fk: R" → R be convex functions. Consider the function f defined by f(x)= E-a;f;(x), where a ; > 0 for j = 1, 2,..., k. Show that f is convex. State and prove a similar result for concave functions. [3.9] Let fi, f2,., fk: R" → R be convex functions. Consider the function f defined by f(x) = max{fj (x), fz(x),..., fx (x)}. Show that f is convex. State and prove a similar result for concave functions.
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