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 .

**References**

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

Gradshteyn, I. S. and Ryzhik, I. M. *Tables of Integrals, Series, and Products, 5th ed.* San Diego, CA:
Academic Press, p. 1100, 1980.

© 1996-9

1999-05-25