Interior extremum theorem

A differentiable function with a local maximum or minimum at an interior point of its domain has derivative zero there. For functions of several variables, the gradient must vanish.

Fermat's theorem on stationary points: A differentiable function has derivative zero at an interior local extremum. This is the theorem’s one-variable form.
Derivative: The instantaneous rate of change of a function, defined through a limit of difference quotients. The theorem’s one-dimensional conclusion is stated in terms of this quantity.
Critical point: A point where a function’s derivative is zero or undefined, or where it lies on a domain boundary. The theorem directs optimization searches toward interior points with zero derivative.
Nondifferentiable extremum: A local maximum or minimum at a point where a function is not differentiable. A sharp corner can be an extremum without a derivative equal to zero.
Directional derivative: The rate at which a function changes at a point along a specified direction. At an interior extremum, opposite directional changes force every directional derivative to vanish.
Differentiability: The property of having a well-defined linear approximation at a point or throughout a region. Differentiability supplies the local rate of change whose value the theorem constrains.
First derivative test: A method that classifies local extrema by examining derivative signs around a critical point. It determines whether a candidate identified by the theorem is actually an extremum.
Subgradient: A generalized derivative represented by a set of slopes for certain nonsmooth functions. Nonsmooth optimization uses generalized conditions when ordinary derivatives fail.
Gradient: A vector of partial derivatives that points in the direction of greatest local increase. Its vanishing expresses the necessary condition for an extremum in several variables.
Local maximum: A point where a function’s value is at least as large as nearby values. The theorem applies when this condition holds at an interior point.

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Maximum and minimumRelated: It explains why stationary points are candidates for differentiable interior extrema.