Can a concave function be convex?
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
Can a set be both convex and concave?
That is, a function is both concave and convex if and only if it is linear (or, more properly, affine), taking the form f(x) = α + βx for all x, for some constants α and β. Economists often assume that a firm’s production function is increasing and concave.
Can a concave function be quasi convex?
The notion of quasiconcavity is weaker than the notion of concavity, in the sense that every concave function is quasiconcave. Similarly, every convex function is quasiconvex. A concave function is quasiconcave.
Why is a linear function both convex and concave?
A linear function will be both convex and concave since it satisfies both inequalities (A. 1) and (A. 2). A function may be con- vex within a region and concave elsewhere.
Can a function be both quasi concave and quasi convex?
Definition and properties A quasilinear function is both quasiconvex and quasiconcave. The graph of a function that is both concave and quasiconcave on the nonnegative real numbers.
What is convex set and concave set?
Let f be a function of many variables, defined on a convex set S. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph; f is convex if the line segment joining any two points on the graph is never below the graph.
How to find if a function is concave or convex?
Let f: S → R where S is non empty convex set in R n, then f ( x) is said to be strictly concave on S if f ( λ x 1 + ( 1 − λ) x 2) > λ f ( x 1) + ( 1 − λ) f ( x 2), ∀ λ ∈ ( 0, 1). A linear function is both convex and concave. f ( x) = | x | is a convex function.
What is the argument for a convex function?
The argument for a convex function is symmetric. Another characterization of a concave function is the following generalization of Jensen’s inequality for functions of a single variable. i=1 λ i = 1. The function f of many variables defined on the convex set S is convex if and only if for all n ≥ 2
What is a convex set?
A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex . . . are convex sets.
How to find if a set is strictly convex?
Let f: S → R where S is non empty convex set in R n, then f ( x) is said to be strictly convex on S if f ( λ x 1 + ( 1 − λ) x 2) < λ f ( x 1) + ( 1 − λ) f ( x 2), ∀ λ ∈ ( 0, 1).