How do you prove X is convex?

If you know calculus, take the second derivative. It is a well-known fact that if the second derivative f (x) is ≥ 0 for all x in an interval I, then f is convex on I. On the other hand, if f(x) ≤ 0 for all x ∈ I, then f is concave on I.

How do you prove that two variables are convex?

If two functions f and g are convex, then so is any weighted combination a f + b g with non- negative coefficients a and b. Likewise, if f and g are convex, then the function max{f,g} is convex. A strictly convex function will have only one minimum which is also the global minimum.

How do you know if a function is Quasiconcave?

Reminder: A function f is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if f(x) ≥ f(y) then f((1 − λ)x + λy) ≥ f(y). Suppose that the function U is quasiconcave and the function g is increasing.

How do you determine if a polygon is convex or concave?

Every polygon is either convex or concave. The difference between convex and concave polygons lies in the measures of their angles. For a polygon to be convex, all of its interior angles must be less than 180 degrees. Otherwise, the polygon is concave.

How do you prove a function is strictly convex?

(1) The function is strictly convex if the inequality is always strict, i.e. if x = y implies that θf ( x) + (1 − θ)f ( y) > f (θ x + (1 − θ) y). (2) A concave function is a function f such that −f is convex. Linear functions are convex, but not strictly convex.

How is a convex set defined?

A convex set is a set of points such that, given any two points A, B in that set, the line AB joining them lies entirely within that set. A convex set; no line can be drawn connecting two points that does not remain completely inside the set.

How do you prove quasiconcave?

Reminder: A function f is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if f(x) ≥ f(y) then f((1 − λ)x + λy) ≥ f(y). Suppose that the function U is quasiconcave and the function g is increasing. Show that the function f defined by f(x) = g(U(x)) is quasiconcave. Suppose that f(x) ≥ f(y).

Can a convex function be quasiconcave?

The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity.