## Convex Function

A function whose value at the Midpoint of every Interval in its Domain does not exceed the Average of its values at the ends of the Interval. In other words, a function is convex on an Interval if for any two points and in ,

If has a second Derivative in , then a Necessary and Sufficient condition for it to be convex on that Interval is that the second Derivative for all in .

Eggleton, R. B. and Guy, R. K. Catalan Strikes Again! How Likely is a Function to be Convex?'' Math. Mag. 61, 211-219, 1988.