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Questions › DSE › DSE 2015 › Q39
2 marks, −0.666666666666667Mathematics › Concavity, convexity and quasi-concavity › Convex sets and functionsModerateabout 3 min

DSE 2015 Question 39: Concavity, convexity and quasi-concavity

Suppose x1,…,xnx_1,\dots,x_nx1​,…,xn​ are positive and λ1,…,λn\lambda_1,\dots,\lambda_nλ1​,…,λn​ are non-negative with ∑i=1nλi=1\sum_{i=1}^n\lambda_i=1∑i=1n​λi​=1. Then

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Suppose x₁,…,xₙ are positive and λ₁,…,λₙ are non-negative with Σᵢ₌₁ⁿλᵢ=1. Then (A) Σᵢ₌₁ⁿλᵢxᵢ≥ x₁^(λ₁)⋯ xₙ^(λₙ) (B) Σᵢ₌₁ⁿλᵢxᵢ< x₁^(λ₁)⋯ xₙ^(λₙ) (C) Σᵢ₌₁ⁿλᵢxᵢ≤√(x₁^(λ₁)⋯ xₙ^(λₙ)) (D) none of the above is necessarily true

Also written as: Suppose x_1,...,x_n are positive and lambda_1,...,lambda_n are non-negative with sum_(i=1)^nlambda_i=1. Then

Concavity, convexity and quasi-concavity has come up 39 times in 6 exams since 2006.

What this question tests

This question is on Convex sets and functions, part of Concavity, convexity and quasi-concavity in Mathematics.

Sets and functions with no dents, and the inequalities that define them.

TestStop has 14 past-paper questions on this subtopic.

Practise Concavity, convexity and quasi-concavityMore on Convex sets and functions

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