Consider a convex function f : R → R. Recall that f is convex if f(tx+ (1-t)y)stf(x)+(1-t)f(y).Also, recall that if f(x) E C2, then an equivalent condition is that f"(x) 2 0.• Prove that for any nf (a1x1 + a2x2 ++ anXn) < a1f (*1) + a2f(x2)+ · .+ anf (an)An.where a1 + a2 + · · ·+ An1, where a; 0.• Prove that – log(x) is convex

Since you have asked multiple question, we will solve the first question for you. If youwant any…

Consider a convex function f : R → R. Recall that f is convex if f(tx + (1– t)y) 0. • Prove that for any n f (a1x1+ a2x2 + +an&n} < aif(¤1}+ a2f(x2) + … + anf(xn} where a1 + a2 + · · ·+ an = 1, where a; > 0.

Consider a convex function f : R → R. Recall that f is convex if f(tx + (1– t)y) 0. • Prove that for any n f (a1x1+ a2x2 + +an&n} < aif(¤1}+ a2f(x2) + … + anf(xn} where a1 + a2 + · · ·+ an = 1, where a; > 0.

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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Consider a convex function f : R → R. Recall that f is convex if f(tx+ (1-t)y)stf(x)+(1-t)f(y).
Also, recall that if f(x) E C2, then an equivalent condition is that f"(x) 2 0.
• Prove that for any n
f (a1x1 + a2x2 +
+ anXn) < a1f (*1) + a2f(x2)+ · .
+ anf (an)
An.
where a1 + a2 + · · ·+ An
1, where a; 0.
• Prove that – log(x) is convex

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