How do you know if a function is quasi convex?
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How do you know if a function is quasi convex?
The function f of many variables defined on a convex set S is quasiconvex if every lower level set of f is convex. (That is, Pa = {x ∈ S: f(x) ≤ a} is convex for every value of a.) Note that f is quasiconvex if and only if −f is quasiconcave.
Is the max of a convex function convex?
A function is convex if and only if the area above its graph is convex. But then, the region above h(x)=max{f(x),g(x)} is the intersection of the area above f and the region above g. Moreover, intersection of convex sets is convex, and that concludes the proof.
Is the MAX operator convex?
It is convex. Recall that a function is convex if and only if, for every in and every , .
How do you show quasi-concave?
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).
Is the minimum of convex functions convex?
Also, the minimum of two convex functions isn’t convex, even though min looks a lot like max. Then h(x) = g(f(x)) is convex. (If f is strictly convex and g is strictly increasing—when x1 < x2, g(x1) < g(x2)—then h is strictly convex as well.)
How do you prove ex is convex?
Convex: see the following figure. 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….3.3 Exercises on concave and convex functions of many variables.
| Π(w’, p’) | ≥ p’f(x*) − w’x* |
|---|---|
| Π(w”, p”) | ≥ p”f(x*) − w”x*. |
What is a quasi concave function?
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form. is a convex set.
What is a concave and convex function?
A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point. Symmetrically, a function of a single variable is convex if every line segment joining two points on its graph does not lie below the graph at any point.