A derivative of zero identifies a stationary location, but it does not by itself prove a maximum or minimum. The function may increase through that point or otherwise fail to change in the required way. Candidate-point analysis therefore treats the derivative equation as a way to locate possibilities, followed by function-value comparisons and, when appropriate, use of the First Derivative Theorem.
Points where the derivative is undefined can matter because extrema do not require a finite derivative at the relevant location. A sharp feature or other nondifferentiable point can still produce a meaningful high or low value. Including these points prevents an optimization procedure from overlooking possible extrema that derivative-zero tests alone cannot detect.
Candidate points supply locations whose function values can be examined for possible local or absolute behavior. A local extremum describes behavior near one point, whereas an absolute extremum reflects comparison across the relevant domain. Evaluating and comparing values at all required candidates, including endpoints when applicable, provides the basis for deciding which type of extremum occurs.
First identify the function’s domain and determine where its derivative is zero or undefined. Next include any domain endpoints that must be considered. Evaluate the function at these locations, then compare the resulting values or examine the surrounding behavior. This sequence turns a list of possible locations into a justified conclusion about maxima and minima.
In constraint-based optimization, candidate-point analysis helps organize the search for optimal values within the allowed domain. The function is examined at relevant interior locations, nondifferentiable locations, and endpoints created by the constraints. Comparing outcomes at these locations supports a defensible choice of the best or worst value rather than relying on a single derivative calculation.
The Extreme Value Theorem provides mathematical context for examining where a function may attain extreme values. Candidate-point analysis complements that perspective by identifying interior derivative conditions, undefined-derivative locations, and endpoints that require direct attention. Together, these ideas support systematic comparisons and help justify conclusions about absolute extrema in calculus and applied mathematics.