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2. Consider a convex function ƒ :R +R. Recall that f is convex if f(tx+(1–t)y) 0. • Prove that for any n f (a1x1 + a2x2 + ...+ anxn) < a1f(x1) + a2f(x2) + ·.+ anƒ(xn) where a1 + az2 + + an = 1, where a; > 0.

2. Consider a convex function ƒ :R +R. Recall that f is convex if f(tx+(1–t)y) 0. • Prove that for any n f (a1x1 + a2x2 + ...+ anxn) < a1f(x1) + a2f(x2) + ·.+ anƒ(xn) where a1 + az2 + + an = 1, where a; > 0.

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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2. Consider a convex function f: R R. Recall that f is convex if f(tx+ (1-t)y) <tf(x)+(1 – t)f(y). Also, recall that if f(x) E C2, then an equivalent condition is that f"(x) > 0. • Prove that for any n f (a1x1 + a2x2 + ..+ anxn) < a1f(x1)+ a2f(x2) + ...+ anf(xn) where a1 + az2 +..+ an = 1, where a; > 0. • Prove that – log(x) is convex • Hence or otherwise conclude the Arithmetic Mean-Geoemtric Mean inequality, i.e., if x;'s are non-negative real numbers, then xi + x2 + + xn
Transcribed Image Text:2. Consider a convex function f: R R. Recall that f is convex if f(tx+ (1-t)y) <tf(x)+(1 – t)f(y). Also, recall that if f(x) E C2, then an equivalent condition is that f"(x) > 0. • Prove that for any n f (a1x1 + a2x2 + ..+ anxn) < a1f(x1)+ a2f(x2) + ...+ anf(xn) where a1 + az2 +..+ an = 1, where a; > 0. • Prove that – log(x) is convex • Hence or otherwise conclude the Arithmetic Mean-Geoemtric Mean inequality, i.e., if x;'s are non-negative real numbers, then xi + x2 + + xn
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