
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the interior extremum theorem, also known as Fermat's theorem, is a theorem which states that at the
local extrema of a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, its
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
is always zero. It belongs to the mathematical field of
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
and is named after French mathematician
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
.
By using the interior extremum theorem, the potential extrema of a function
, with derivative
, can found by solving an
equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
involving
. The interior extremum theorem gives only a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for extreme function values, as some stationary points are
inflection points (not a maximum or minimum). The function's
second derivative
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
History
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
proposed in a collection of treatises titled ''Maxima et minima'' a method to find maximum or minimum, similar to the modern interior extremum theorem, albeit with the use of
infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s rather than derivatives.
After
Marin Mersenne
Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
passed the treatises onto
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
, 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:
:Let
be a function from an
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
to , and suppose that
is a point where
has a local extremum. If
is differentiable at
, then
.
Another way to understand the theorem is via the
contrapositive
In logic and mathematics, contraposition, or ''transposition'', refers to the inference of going from a Conditional sentence, conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrap ...
statement: if the derivative of a function at any point is not zero, then there is not a local extremum at that point. Formally:
:If
is differentiable at
, and
, then
is not a local extremum of
.
Corollary
The global extrema of a function ''f'' on a
domain ''A'' occur only at
boundaries, non-differentiable points, and stationary points.
If
is a global extremum of ''f'', then one of the following is true:
* boundary:
is in the boundary of ''A''
* non-differentiable: ''f'' is not differentiable at
* stationary point:
is a stationary point of ''f''
Extension
A similar statement holds for the
partial derivatives of
multivariate functions. Suppose that some real-valued function of the real numbers
has an extremum at a point
, defined by
. If
is differentiable at
, then:
where
.
The statement can also be extended to
differentiable manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
. If
is a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
on a manifold
, then its local extrema must be
critical points of
, in particular points where the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
is zero.
Applications
The interior extremum theorem is central for determining
maxima and minima
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
of
piecewise differentiable functions of one variable: an extremum is either a
stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
(that is, a
zero
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
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 provide the maximum and the minimun, simply by comparing the obtained values.
Proof
Suppose that
is a local maximum. (A similar argument applies if
is a local minimum.) Then there is some
neighbourhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
around
such that
for all
within that neighborhood. If
, then the
difference quotient
In single-variable calculus, the difference quotient is usually the name for the expression
: \frac
which when taken to the Limit of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The ...
is non-positive for
in this neighborhood. This implies
Similarly, if
, then the difference quotient is non-negative, and so
Since
is differentiable, the above limits must both be equal to
. This is only possible if both limits are equal to 0, so
.
See also
*
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
*
Maxima and minima
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
*
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
*
Extreme value
*
arg max
*
Adequality, a term of Fermat's related to his method of finding extrema, and the subject of some controversy among mathematical historians
References
External links
*
*
{{DEFAULTSORT:Fermat's Theorem (Stationary Points)
Theorems in real analysis
Differential calculus
Articles containing proofs
Theorems in calculus