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When I took calculus, we didn't use "concave" and "convex" - rather, we (and the AP exam) used "concave up" and "concave down." I still use these as a grad student. One can also remember that concave functions look like the opening of a cave.
So in terms of getting a unique (or at least convex) set of solutions to the FOC, concavity is 'global', whereas quasi-concavity is only 'local'. $\endgroup$ – Pete Caradonna Commented Nov 1, 2017 at 17:29
A function f: R → R is convex (or "concave up") provided that for all x, y ∈ R and t ∈ [0, 1], f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y). Equivalently, a line segment between two points on the graph lies above the graph, the region above the graph is convex, etc. I want to know why the word "convex" goes with the inequality in this ...
Dec 24, 2018. #35. Mistershine. said: Unless you have some weird medical condition that causes you to have concave thumbs I don't understand how convex sticks are comfy. Concave is better if you press and drag the analog sticks to move them. Convex is better if you push the analog sticks with your thumb to move them.
While I appreciate Kajelad's thoroughness, there's a much simpler way to look at this if you are simply trying to confirm whether the function is convex or concave or neither.
How to determine whether a function is concave, convex, quasi-concave and quasi-convex. 0.
A strictly convex function will always take a unique minimum. For a convex function which is not strictly convex the minimum needs not to be unique. For example, f(x) above takes its minimum everywhere between -4 and 4. Hence, the minimum is not unique. For g(x) = x2, however, there is only one unique minimum at x = 0. Share.
2. It depends on your definition of concave: there are the notion of "concave" and "strictly concave". In x ≥ 0 x ≥ 0 arctan(x) arctan (x) is concave, but not strictly concave. (The difference between the two notions translate in terms of the second derivative as the two conditions f′′ ≤ 0 f ″ ≤ 0 or f′′ <0 f ″ <0) – Dario.
$$ \text{convex}\implies\text{pseudoconvex}\implies\text{strictly quasiconvex}\implies\text{quasiconvex} $$ An expository paper that is a great starting point for learning about these functions is given below.
Mar 6, 2014 at 22:17. 3. That's great and actually does not rely on derivative arguments: A function f f is convex if for every x x there exists c c such that for all y y it holds f(y) ≥ f(x) + c(y − x) f (y) ≥ f (x) + c (y − x). You showed that c =ex c = e x works. This does not only show that f f is convex but also that ex e x is a ...