(a) Let n be a positive integer, and let t₁,1,..., tn be positive numbers satisfying t₁+...+tn = 1.If I is an interval and f: I→ R is concave down, show that for any ₁,...,n in I,+ tnxn is in I, andt₁x₁ +f(t₁x₁ +... + tnxn) ≥ t₁f(x₁) +...+ tnf(xn).(b) The arithmetic mean geometric mean inequality says that if a₁, ..., an are n positive num-bers, thena₁ + ... + ann> va₁...an.Use the concavity of a suitable function to prove this inequality. You need to justify all thestatements.

Given n be a positive integer and t1,...tn be positive numbers satisfying t1+......tn=1We need to…

(a) Let n be a positive integer, and let t₁, ..., tn be positive numbers satisfying t₁+...+tn = 1. If I is an interval and f: IR is concave down, show that for any ₁,...,n in I, t₁x₁ + ...+ tnxn is in I, and f(t₁x₁ + + tnxn) ≥tif(x₁) + + tnf(xn). (b) The arithmetic mean geometric mean inequality says that if a₁, ..., an are n positive num- bers, then statements. a₁ + + an n > a₁...an. Use the concavity of a suitable function to prove this inequality. You need to justify all the

(a) Let n be a positive integer, and let t₁, ..., tn be positive numbers satisfying t₁+...+tn = 1. If I is an interval and f: IR is concave down, show that for any ₁,...,n in I, t₁x₁ + ...+ tnxn is in I, and f(t₁x₁ + + tnxn) ≥tif(x₁) + + tnf(xn). (b) The arithmetic mean geometric mean inequality says that if a₁, ..., an are n positive num- bers, then statements. a₁ + + an n > a₁...an. Use the concavity of a suitable function to prove this inequality. You need to justify all the

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.1: Postulates For The Integers (optional)

Problem 27E: Let x and y be in Z, not both zero, then x2+y2Z+.

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