AP Calculus AB and BC glossary

Second derivative test

The second derivative test classifies a critical point by evaluating the second derivative there. A negative value means a local maximum, a positive value means a local minimum, and a value of zero makes the test inconclusive.

The logic is about concavity. At a critical point on a concave down curve the graph must turn over, giving a maximum; on a concave up curve it must turn up, giving a minimum.

When $f''$ is zero at the critical point the test says nothing at all, and you must fall back on the first derivative test. Compare $x^4$ , $-x^4$ , and $x^3$ , which all have $f'(0) = f''(0) = 0$ but a minimum, a maximum, and neither.

The mistake

Concluding a point is neither a maximum nor a minimum when $f''$ is zero. Inconclusive means you need a different test, not that no extremum exists.

Appears in: Unit 5: Analytical Applications