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Convexity of a Function 
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#1
Apr1207, 02:33 PM

P: 117

What does it mean for a function to be convex (or concave) on an interval [a,b]? I understand what a function is and what an interval is, but I don't get what "convexity" is.



#2
Apr1207, 02:47 PM

P: 998

A (twicedifferentiable) function is concave at a point if its second derivative is negative at that point. Similarly a function is convex at a point if its second derivative is positive at that point.
You can extend the definition to functions that aren't differentiable also; see http://en.wikipedia.org/wiki/Concave_function. Intuitively: A concave (or "concave down") function is one that is "cupped" downwards. For example, the parabola [itex]x^2[/itex] is concave throughout its domain, and the parabola [itex]x^2[/itex] is convex throughout its domain. There are functions which are "cupped" but don't actually have the cup shape. For example, [itex]1/x[/itex] is concave on the negative reals and convex on the positive reals, however it doesn't have any extrema at all. Another way to present it is: A function [itex]f[/itex] is convex on an interval if the set of points above its graph on that interval is a convex set; that is, if[itex]p = (x_1, y_1)[/itex] and [itex]q = (x_2, y_2)[/itex] are points with [itex]x_1, x_2[/itex] on the interval of interest, [itex]y_1 \geq f(x_1)[/itex], and [itex] y_2 \geq f(x_2)[/itex], then the straight line joining [itex]p[/itex] to [itex]q[/itex] lies entirely above the graph of [itex]f[/itex]. Then you can define [itex]f[/itex] is concave whenever [itex]f[/itex] is convex. 


#3
Apr1207, 03:29 PM

P: 53

A convex set is a set where all points can be connected with a straight line inside the set (so every point can "see" every other). A function is convex if the set above it (ie the set {(x,y):y>f(x)}) is convex.
If the function is twice differentiable, this is equivalent with that the second derivative is everywhere nonnegative. 


#4
Apr1207, 03:32 PM

P: 117

Convexity of a Function



#5
Apr1207, 05:19 PM

P: 53

Yes. (damn character limit)



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