AP Calculus AB and BC glossary

Relative maximum

Also called: Local maximum

A relative maximum is a value that is larger than every nearby value of the function. It occurs at a critical number where the derivative changes from positive to negative, meaning the graph stops rising and starts falling. It is a local peak and need not be the largest value overall.

You confirm a relative maximum one of two ways. The first derivative test looks for f′f' switching from positive to negative at the critical number. The second derivative test checks that f′(c)=0f'(c) = 0 and f′′(c)<0f''(c) < 0, so the curve is concave down there.

Relative means local. A function can have several relative maxima at different heights, and none of them has to be the absolute maximum. On a closed interval the absolute maximum might instead sit at an endpoint, which is never a relative extremum.

The mistake

Claiming a relative maximum wherever f′(c)=0f'(c) = 0. The derivative must actually change sign. For f(x)=x3f(x) = x^3 the derivative is zero at the origin, yet there is no maximum, because f′f' stays positive on both sides.

Appears in: Unit 5: Analytical Applications