Concept preview

What you will be able to do

1. 01Define concavity: a function is concave up where $f''(x)>0$ (its slopes are increasing and the graph cups upward like a bowl) and concave down where $f''(x)<0$ (slopes decreasing, cupping downward like a dome).
2. 02Build and read a sign chart of $f''$ to determine the intervals of concavity, and identify inflection points as the places where $f''$ CHANGES sign.
3. 03Distinguish an inflection point from a mere zero of $f''$, and distinguish concavity (the sign of $f''$) from increasing/decreasing behaviour (the sign of $f'$).

A function is concave up on an interval if its graph lies above its tangent lines and cups upward like the inside of a bowl; it is concave down if the graph lies below its tangent lines and arches over like a dome. The analytic test is clean: on an interval where $f''(x)>0$ the function is concave up, and on an interval where $f''(x)<0$ it is concave down. The reason is that $f''$ is the derivative of $f'$, so $f''>0$ means the slope $f'$ is increasing — the tangent lines are tilting more and more upward as you move right — which is exactly the bowl shape. When $f''<0$, the slope is decreasing and the tangents tilt downward, giving the dome.