Concave Function

A function $f(x)$ is said to be concave on an interval $[a,b]$ if, for any points $x_1$ and $x_2$ in $[a,b]$, the function $-f(x)$ is Convex on that interval. If the second Derivative of $f$

$$f''(x) > 0,$$

on an open interval $(a,b)$ (where $f''(x)$ is the second Derivative), then $f$ is concave up on the interval. If

$$f''(x) < 0$$

on the interval, then $f$ is concave down on it.

References

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