Interior extremum theorem on arvau

Wikipedia / Interior extremum theorem

In calculus and real analysis, the interior extremum theorem states that any local extremum of a real function at which it is differentiable is a stationary point. It is also known as Fermat's theorem, named after the French mathematician Pierre de Fermat.

The interior extremum theorem gives a necessary, but not sufficient condition for local extrema at which the function is differentiable, as some stationary points are not local extrema.

The second derivative, if non-zero, can be used to determine whether a local extremum at which the function is twice differentiable is a maximum or a minimum. However, the second derivative can be zero at local extrema.

History

Pierre de Fermat proposed in a collection of treatises titled Maxima et minima a method to find maximum or minimum, similar to the modern interior extremum theorem using an approach he called adequality. After Marin Mersenne passed the treatises onto René Descartes, Descartes was doubtful, remarking "if [...] he speaks of wanting to send you still more papers, I beg of you to ask him to think them out more carefully than those preceding". Descartes later agreed that the method was valid.

Statement

One way to state the interior extremum theorem is that, if a function has a local extremum at some point and is differentiable there, then the function's derivative at that point must be zero. In precise mathematical language:

be a function from an open interval to .

Another way to understand the theorem is via the contrapositive statement: if the derivative of a function at any point is not zero, then there is not a local extremum at that point. Formally:

Corollary

Every global extremum of a function f on a domain A occurs only at the boundary of A is a global extremum of f, then one of the following is true:

boundary: is in the boundary of A
non-differentiable: f
stationary point:

The function has no extrema. The function has no global extrema, although it has local extrema.

Extension

A similar statement holds for the partial derivatives of multivariate functions.

The statement can also be extended to differentiable manifolds is a differentiable function, then its local extrema must be critical points, in particular points where the exterior derivative is zero.

Applications

The interior extremum theorem is central for determining maxima and minima of piecewise differentiable functions of one variable: an extremum is either a stationary point (that is, a zero of the derivative), a non-differentiable point (that is a point where the function is not differentiable), or a boundary point of the domain of the function. Since the number of these points is typically finite, the computation of the values of the function at these points provides the maximum and the minimum, simply by comparing the obtained values.

Proof

is a local minimum.) Then there is some neighbourhood, then the difference quotient.