This demonstration illustrates the concept of concavity. The points $a$ and $b$ can be dragged along the $x$-axis. The corresponding points $(a,f(a))$ and $(b,f(b))$ are joined by a line segment. The midpoint of this line segment is $\left(\frac{a+b}{2},\frac{f(a)+f(b)}{2}\right)$. If a function $f(x)$ is concave up in a region containing $[a,b]$ then the average value of $f(a)$ and $f(b)$ is at least as big as the value of $f$ halfway between $a$ and $b$. In other words, $\frac{f(a)+f(b)}{2} \ge f\left(\frac{a+b}{2}\right)$.