AP Calculus AB and BC glossary

Concavity test

The concavity test says a graph is concave up on any interval where the second derivative is positive and concave down on any interval where the second derivative is negative. It describes intervals, unlike the second derivative test, which classifies a single critical point.

The test is the increasing/decreasing test applied to $f'$ . Concave up means the slope is increasing, so the same sign chart machinery works with $f''$ in place of $f'$ , using the zeros and undefined points of $f''$ as the dividers.

Two tests use $f''$ for different jobs. The concavity test reads the sign of $f''$ across an interval and returns a shape; the second derivative test evaluates $f''$ at a single critical number $c$ where $f'(c) = 0$ and returns a classification whenever $f''(c)$ is nonzero, while a value of zero leaves it inconclusive. A sign chart answers the first, a single substitution usually settles the second, and the answers are not interchangeable.

The mistake

Treating $f''(c) > 0$ as evidence of a minimum at $c$ . That conclusion belongs to the second derivative test and only follows once $f'(c) = 0$ is established. The function $f(x) = e^{x}$ has $f''(x) > 0$ everywhere and no minimum anywhere.

Appears in: Unit 5: Analytical Applications