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Let S be a non-empty convex subset of R" and f a real valued concave function on S. Prove the following statement: If x, x' e S maximizes f (over S), then for any a E (0,1), ax +(1- a)x' also maximizes f (over S).

Let S be a non-empty convex subset of R" and f a real valued concave function on S. Prove the following statement: If x, x' e S maximizes f (over S), then for any a E (0,1), ax +(1- a)x' also maximizes f (over S).

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) Let S be a non-empty convex subset of R" and f a real valued concave function on S. Prove the following statement: If x, x' e S maximizes f (over S), then for any a E (0, 1), ax + (1- a)x' also maximizes f (over S). 4) Prove the following statement: For any g : R" R and any f: D R, where D is a convex subset of R that contains the range of g, if g is a concave function and f is a convex and non-increasing function, then fog: R" -R is a convex function. (Note that f and g does not have to be differentiable.)
Transcribed Image Text:3) Let S be a non-empty convex subset of R" and f a real valued concave function on S. Prove the following statement: If x, x' e S maximizes f (over S), then for any a E (0, 1), ax + (1- a)x' also maximizes f (over S). 4) Prove the following statement: For any g : R" R and any f: D R, where D is a convex subset of R that contains the range of g, if g is a concave function and f is a convex and non-increasing function, then fog: R" -R is a convex function. (Note that f and g does not have to be differentiable.)
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