AP Calculus AB and BC glossary
Relative minimum
Also called: Local minimum
A relative minimum is a value that is smaller than every nearby value of the function. It occurs at a critical number where the derivative changes from negative to positive, meaning the graph stops falling and starts rising. It is a local valley and need not be the smallest value overall.
Two tests confirm it. The first derivative test wants changing from negative to positive at the critical number. The second derivative test wants with , so the curve is concave up and holds like a bowl.
A relative minimum is the mirror image of a relative maximum, and a single function can hold several of them at different depths. None is required to be the absolute minimum, which on a closed interval may live at an endpoint instead.
The mistake
Reversing the direction of the sign change. Negative to positive gives a minimum, the valley; positive to negative gives a maximum. Swapping the two is the most common sign-chart slip on free response.
Appears in: Unit 5: Analytical Applications