AP Calculus AB and BC glossary

Critical point

Also called: Critical number, Critical value

A critical point of a function is an interior point of its domain where the derivative is either zero or undefined. Every local maximum and minimum occurs at a critical point, though not every critical point is an extremum.

Both halves of the definition matter. Setting f′(x)=0f'(x) = 0 finds the smooth turning points, but points where f′f' fails to exist, such as the corner of ∣x∣|x| at the origin, are critical too and are the ones most often missed.

Critical points are candidates, not conclusions. The function f(x)=x3f(x) = x^3 has f′(0)=0f'(0) = 0 yet no extremum there, because the derivative does not change sign.

The mistake

Counting a point where the denominator of f′f' vanishes but which is outside the domain of ff. A critical point must be in the domain of the original function.

Appears in: Unit 5: Analytical Applications