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1. math.ucr.edu › ~res › math153-2019Concave and Convex Functions - Department of Mathematics

   math.ucr.edu › ~res › math153-2019

   Theorem 3. Let C R be an open interval. 1. f: C!R is concave i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a: For strict concavity, the inequalities are strict. 2. f: C!R is convex i for any a;b;c2C, with a<b<c, f(b) f(a) b a f(c) f(b) c b; and, f(b) f(a) b a f(c) f(a) c a:

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3. pareto.uab.cat › xvila › Micro_INotes on Concavity, Convexity, Quasiconcavity and Quasiconvexity

   pareto.uab.cat › xvila › Micro_I

   Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.

4. www.math.lamar.edu › faculty › maesumiConcavity - Lamar.edu

   www.math.lamar.edu › faculty › maesumi

   The following notions of concavity are used to describe the increase and decrease of the slope of the tangent to a curve. Concavity If the function f(x) is differentiable on the interval a x b, then the graph of f is. concave upward on a x b if f is increasing on the interval concave downward on a x b if f is decreasing on the interval.

5. math.gmu.edu › math215 › textbook3.5 Quadratic Approximation and Convexity/Concavity

   math.gmu.edu › math215 › textbook

   The analysis of quadratic functions from Chapter 1 becomes a fundamental tool for describing behavior that is beyond the linear approximation, such as bending (convexity/concavity).

6. cdn.kutasoftware.com › Worksheets › Calc04 - Intervals of Concavity - Kuta Software

   cdn.kutasoftware.com › Worksheets › Calc

   Intervals of Concavity Date_____ Period____ For each problem, find the x-coordinates of all points of inflection, find all discontinuities, and find the open intervals where the function is concave up and concave down.

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7. faculty.fiu.edu › ~boydj › mathecon21. Concave and Quasiconcave Functions

   faculty.fiu.edu › ~boydj › mathecon

   The function f(x) = ex is convex, while f(x) = lnx is concave. It’s also possible to have flat spots in the graph of a convex or concave function. Figure 21.1.3: This function is convex, but not strictly convex. The flat portions of the graph rule out strict convexity.

8. People also ask

   How do you prove a concave function?

   • Any afine function is both concave and convex. Proof. The proof follows from Theorem 6 above and from the fact that f(x) xi, f(x) −xi, and f(x) Theorem 9. Let f : S ⊂ Rn → R be a concave (convex) function, and let g : R → R be concave (convex) and increasing. Then (f ◦ g) : S ⊂ Rn → R is a concave (convex) function. Proof.

   Notes on Concavity, Convexity, Quasiconcavity and Quasiconvexity

   pareto.uab.cat/xvila/Micro_I/ConcavityConvexityQuasiconcaviyQuasiconvexity.pdf

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   Can convex and concave functions be collected together without proof?

   • Several easily established properties of convex and concave functions are collected together without proof in Theorem 21.1.4. Theorem 21.1.4. A function f is (strictly) convex if and only if −f is (strictly) concave. A positive scalar multiple of a concave (convex) function is concave (convex).

   21. Concave and Quasiconcave Functions - Florida International Univer…

   faculty.fiu.edu/~boydj/mathecon/math21.pdf

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   What are convex and concave functions?

   • Convex and Concave Functions. Let S be a convex set. tx + (1 − t)y ≤ tf(x) + (1 − t)f(y). tx + (1 − t)y ≥ tf(x) + (1 − t)f(y). strictly convex or strictly concave. One consequence is that for convex functions, every chord of the graph of the function lies on or above the graph.

   21. Concave and Quasiconcave Functions - Florida International Univer…

   faculty.fiu.edu/~boydj/mathecon/math21.pdf

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   How do you analyze concavity if n 1 is separable?

   • the de nition of concavity.special case in which N > 1 is e ectively as easy to analyze as N = 1 is when the function is separable in the sense that it is the sum of univariate functions. For example, the function h(x1; 2) = ex1 + ex2 is separable. More generally, suppose Cn ar intervals a d let C = QCn. Then h : C ! R is separable i

   Concave and Convex Functions | Department of Mathematics

   math.ucr.edu/~res/math153-2019/concavity.pdf

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   What is a concavity tangent?

   • The following notions of concavity are used to describe the increase and decrease of the slope of the tangent to a curve. In Figure 3.11, for example, the production curve was concave upward to the left of the point of diminishing returns and concave downward to the right of this point.

   Concavity - Lamar.edu

   www.math.lamar.edu/faculty/maesumi/applied calculus/hoffman/ch03sec02.pdf

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   How do you know if a function is concave (convex)?

   • A function f is (strictly) convex if and only if −f is (strictly) concave. A positive scalar multiple of a concave (convex) function is concave (convex). The sum of two concave (convex) functions is concave (convex). If φ is concave (convex) and weakly increasing on R and f is a concave (convex) function, then φ ◦ f is concave (convex).

   21. Concave and Quasiconcave Functions - Florida International Univer…

   faculty.fiu.edu/~boydj/mathecon/math21.pdf

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9. math.gmu.edu › ~skim22 › 2010S Math1083.2. Concavity and Points of Inflecti - George Mason University

   math.gmu.edu › ~skim22 › 2010S Math108

   Definition. If f (x) is differentiable on the interval a < x < b, then the graph of f is. concave upward on a < x ′ < b if f is increasing on the interval. concave downward on a < x < b if f ′ is decreasing on the interval.