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.
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.
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 non-negative.