AP Calculus AB and BC glossary

Critical number

A critical number is an x-value in the domain of a function where its derivative equals zero or fails to exist. Critical numbers are the only candidates for a relative extremum, but a candidate is not a guarantee. The critical number is the x-value itself, not the point on the graph.

The critical number is the input. Pair it with the output and you get the critical point $(c, f(c))$ , an ordered pair on the graph. AP questions ask for one or the other, so read whether they want the location $c$ or the full coordinates.

Two situations produce a critical number: $f'(c) = 0$ at a smooth turning point, or $f'(c)$ undefined at a corner or vertical tangent. The second kind, like the corner of $f(x) = |x|$ at $x = 0$ , is the one students forget to list.

The mistake

Counting an $x$ -value where $f'$ blows up but $f$ itself is undefined. A critical number must lie in the domain of the original function, so $x = 0$ for $f(x) = 1/x$ does not qualify.

Appears in: Unit 5: Analytical Applications