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- Every convex function is quasiconvex. A concave function can be quasiconvex. For example, x ↦ log (x) {displaystyle xmapsto log (x)} is both concave and quasiconvex. Any monotonic function is both quasiconvex and quasiconcave.
en.wikipedia.org/wiki/Quasiconvex_function
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A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$. Another representation of quasi-concave functions: $f(x)$ is called quasi- concave if and only if
For example, f(x) transformation, but g(f(x)) = = −x2 is concave, and g(x) 2 = ex is a monotonic e−x2. 2 is not concave. This is problematic when we want to analyze things like utility which we consider to be ordinal concepts. A weaker condition to describe a function is quasiconvexity (or quasiconcav-ity).
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A function is concave (convex) if the graph of the function is always above (below) any chord (line segment between two points in the graph). Remark 4. f concave ⇔−f convex. Example 5. Let S = [0,∞) and consider f(x) = √ x and g(x) = −f(x) = − √ x f is a concave function and g is a convex function. 1.2 Selected properties of ...
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. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints.
• The function f is quasiconvex if for all x,y ∈ C and 0 ⩽ λ ⩽ 1, we have f ((1−λ)x+λy) ⩽ max{f(x),f(y)}. • The function is quasiconcave if for all x,y ∈ C and 0 ⩽ λ ⩽ 1, we have f ((1−λ)x+λy) ⩾ min{f(x),f(y)}. 7.2.2 Exercise Prove the following. 1. Every convex function is quasiconvex. Every concave function is ...
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Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
Yes! Both linear and affine functions are both convex and concave. Example 21.1.2: Convex Functions. Any linear function f(x) = p·x is both concave and convex, as is the generic affine function f(x) = a+p·x.
