Maximum and minimum

A maximum is the greatest value attained by a function or set, and a minimum is the least. They may be global over a domain or local within a neighborhood.

Supremum and infimum: The least upper bound and greatest lower bound of a set, whether or not either belongs to the set. A supremum or infimum can exist even when a function never attains its maximum or minimum.
Fermat's theorem on stationary points: A theorem stating that a differentiable function has derivative zero at any interior local extremum. It explains why stationary points are candidates for differentiable interior extrema.
Mathematical optimization: The study of choosing variables to minimize or maximize an objective under constraints. Maximum and minimum values are the central outcomes sought in optimization.
Monotonic function: A function that consistently increases or consistently decreases as its input increases. A strictly monotonic function on an interval has no interior local extrema.
Local extremum: A maximum or minimum relative to values in a neighborhood of a point. Local extrema capture nearby peaks and valleys without requiring the greatest or least value globally.
First derivative test: A method that classifies a critical point by checking the sign change of a function's derivative. A change from increasing to decreasing identifies a local maximum; the reverse identifies a minimum.
Linear programming: Optimization of a linear objective over a region defined by linear constraints. Its finite optima, when they exist, occur at vertices of the feasible region.
Saddle point: A stationary point that is neither a local maximum nor a local minimum. It shows why a zero derivative alone cannot establish an extremum.
Global extremum: A maximum or minimum that bounds every value in the specified domain or region. Global extrema compare the entire domain rather than only nearby points.
Second derivative test: A test using a function's second derivative at a stationary point to classify its local behavior. Positive curvature indicates a local minimum and negative curvature a local maximum under the test's conditions.

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