What is the relation between convexity and quasi convexity?
For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. 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.
How do you test for quasi concavity?
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).
Does composition preserve convexity?
Composition: this is a bit tricky, as composition rules that preserve convexity (or concavity) rely on monotonicity conditions. Here are a few results to remember, in the setting f = h ◦ g, where g : Rn → R, h : R → R, so f : Rn → R.
What is the difference between concave and quasiconcave?
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. A convex function is quasiconvex.
Is quasi concavity ordinal?
therefore the upper contour set remains convex Thus quasiconcavity is an ordinal property.
Is e x convex?
The function ex is differentiable, and its second derivative is ex > 0, so that it is (strictly) convex. Hence by a result in the text the set of points above its graph, {(x, y): y ≥ ex} is convex.
Is the composition of 2 convex functions convex?
No, as any norm, norm is homogeneous, in particular, so it is not strictly convex.
Is the composition of 2 convex functions itself a convex function?
In general, the answer is negative.
Can a function be both concave and convex?
Absolutely ! Take a look at a function that is both convex and concave on . A simple example of such a function is given by all the linear functions : is a perfectly fit example of a function that is both convex and concave.
What does 2nd derivative tell you?
The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing.