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Questions › DSE › DSE 2019 › Q17
0 marksMathematics › Concavity, convexity and quasi-concavity › Testing concavity and convexityHardabout 3 min

DSE 2019 Question 17: Concavity, convexity and quasi-concavity

Consider a function f:ℜ2→ℜf:\Re^2\to\Ref:ℜ2→ℜ. Suppose that for every p∈ℜ2p\in\Re^2p∈ℜ2 there exists x(p)∈ℜ2x(p)\in\Re^2x(p)∈ℜ2 such that f(x(p))≥1f(x(p))\ge 1f(x(p))≥1 and p⋅x(p)≤p⋅yp\cdot x(p)\le p\cdot yp⋅x(p)≤p⋅y for every y∈ℜ2y\in\Re^2y∈ℜ2 with f(y)≥1f(y)\ge 1f(y)≥1. Define g:ℜ2→ℜg:\Re^2\to\Reg:ℜ2→ℜ by g(p)=p⋅x(p)g(p)=p\cdot x(p)g(p)=p⋅x(p). Then ggg is:

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Consider a function f:ℜ²→ℜ. Suppose that for every p∈ℜ² there exists x(p)∈ℜ² such that f(x(p))≥ 1 and p· x(p)≤ p· y for every y∈ℜ² with f(y)≥ 1. Define g:ℜ²→ℜ by g(p)=p· x(p). Then g is: (A) linear (B) quasi-convex (C) convex (D) concave

Also written as: Consider a function f: Re^2-> Re. Suppose that for every p in Re^2 there exists x(p) in Re^2 such that f(x(p))>= 1 and p* x(p)<= p* y for every y in Re^2 with f(y)>= 1. Define g: Re^2-> Re by g(p)=p* x(p). Then g is:

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

What this question tests

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

Checking concavity or convexity from a definition, derivatives or operations that preserve it.

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