Second derivative test

A criterion for classifying a stationary point of a twice-differentiable function from the sign of its second derivative there. A positive value indicates a local minimum, and a negative value indicates a local maximum.

Taylor's theorem: A theorem that represents a smooth function near a point by a polynomial approximation with a controlled remainder. Its local expansion explains why the quadratic term determines behavior when the first derivative vanishes.
Derivative: A quantity describing the instantaneous rate of change of a function with respect to its input. The test uses the second derivative, which measures how the first derivative changes.
Unconstrained optimization: Optimization of an objective function without equality or inequality constraints. The test provides a local classification step after finding stationary candidates.
First derivative test: A method that classifies extrema by checking how a function's derivative changes sign. It can classify a stationary point even when the second derivative is zero.
Local extremum: A point where a function's value is no greater or no less than nearby values. The test distinguishes the two kinds of local extremum at a stationary point.
Second derivative: The derivative of a function's first derivative. Its value at the stationary point supplies the test's curvature signal.
Critical point: A point where a function's derivative is zero or undefined. Critical points are candidates for extrema, though the test applies only where the required derivatives exist.
Second-order sufficient condition: A curvature condition that guarantees a strict local optimum in constrained or multivariable optimization. It extends second-derivative reasoning to optimization settings beyond a single-variable function.
Concavity: A property describing whether a function bends upward or downward over an interval. Positive and negative second derivatives encode upward and downward bending near the point.
Stationary point: A point where a differentiable function's derivative is zero. The test applies at such points, where first-order change vanishes.

Linked from 8 pages