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Izzhov
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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.
Data said:A (twice-differentiable) 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.
Convexity of a function refers to the property of a mathematical function where the line segment between any two points on the function always lies above or on the function itself. In other words, a function is convex if its graph is always curved upwards or flat, never bending downwards.
Mathematically, a function f(x) is convex if for any two points x1 and x2 in its domain, and for any value of λ between 0 and 1, the following inequality holds: f(λx1 + (1-λ)x2) ≤ λf(x1) + (1-λ)f(x2). This is known as the convex combination property.
Convex functions play a crucial role in optimization problems, as they have a unique global minimum that can be easily found using various methods. This makes it easier to find the optimal solution in various applications, such as in economics, engineering, and machine learning.
No, a function cannot be both convex and concave. A function is convex if the line segment between any two points on the function lies above or on the function, while a concave function has the line segment between any two points lying below or on the function. Therefore, a function can only be either convex or concave, but not both.
There are various methods to test for convexity in a function, such as checking the second derivative of the function, using the convex combination property, or graphically plotting the function and checking for convexity visually. Additionally, there are also specific conditions and rules that can help determine convexity, depending on the type of function.