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{{short description|Method to find local maxima and minima of differentiable functions on open sets}}
{{About|Fermat's theorem concerning the maximums and minimums of functions|other theorems also named after Pierre de Fermat|Fermat's theorem}}
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[[File:Interior extremum graph.svg|thumb|A differentiable function graph with lines tangent to the minimum and maximum. Fermat's theorem guarantees that the [[slope]] of these lines will always be zero.]]
In [[mathematics]], '''Fermat's theorem''' (also known as '''interior [[Maximum and minimum|extremum]] theorem''') is a theorem which states that at the local extrema of a [[differentiable function]], its [[derivative]] is always zero. It belongs to the mathematical field of [[real analysis]] and is named after French mathematician [[Pierre de Fermat]].

By using Fermat's theorem, the potential extrema of a function <math>f</math>, with derivative <math>f'</math>, can found by solving an [[equation]] involving <math>f'</math>. Fermat's theorem gives only a [[necessary condition]] for extreme function values, as some stationary points are [[inflection point]]s (not a maximum or minimum). The function's [[second derivative]], if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.

== 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 Fermat's theorem, albeit with the use of [[infinitesimal]]s rather than derivatives.<ref>{{Cite book |last=Fikhtengol'ts |first=G.M. |title=The Fundamentals of Mathematical Analysis |publisher=[[Pergamon Press]] |year=1965 |isbn=978-0-08-013473-4 |doi=10.1016/C2013-0-02242-6}}</ref>{{Rp|page=|pages=456–457}}<ref name=":0" />{{Rp|page=2}} 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".<ref name=":0" />{{Rp|page=3}} Descartes later agreed that the method was valid.<ref name=":0" />{{Rp|page=8}}

==Statement==
One way to state Fermat's theorem is that, if a function has a local [[Maxima and minima|extremum]] at some point and is [[differentiable function|differentiable]] there, then the function's derivative at that point must be zero. In precise mathematical language:

:Let <math>f\colon (a,b) \rightarrow \mathbb{R}</math> be a function and suppose that <math>x_0 \in (a,b)</math> is a point where <math>f</math> has a local extremum. If <math>f</math> is differentiable at <math>x_0</math>, then <math>f'(x_0) = 0</math>.<ref>{{Cite journal |last=Bronshtein |first=I. N. |last2=Semendyayev |first2=K. A. |date=1972 |title=A Guide Book to Mathematics |url=https://link.springer.com/book/10.1007/978-1-4684-6288-3 |journal=SpringerLink |language=en |doi=10.1007/978-1-4684-6288-3}}</ref>{{Rp|page=377}}

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:
:If <math>f</math> is differentiable at <math>x_0 \in (a,b)</math>, and <math>f'(x_0) \neq 0</math>, then <math>x_0</math> is not a local extremum of <math>f</math>.

===Corollary===
The global extrema of a function ''f'' on a [[Domain of a function|domain]] ''A'' occur only at [[boundary (topology)|boundaries]], non-differentiable points, and stationary points.
If <math>x_0</math> is a global extremum of ''f'', then one of the following is true:<ref name=":0">{{Cite journal |last=Monks |first=Kenneth M |date=February 20, 2023 |title=Fermat’s Method for Finding Maxima and Minima |url=https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf |journal=[[MAA Convergence]]}}</ref>{{Rp|page=1}}
* '''boundary:''' <math>x_0</math> is in the boundary of ''A''
* '''non-differentiable:''' ''f'' is not differentiable at <math>x_0</math>
* '''stationary point:''' <math>x_0</math> is a stationary point of ''f''

===Extension===

In higher dimensions, exactly the same statement holds; however, the proof is slightly more complicated. The complication is that in 1 dimension, one can either move left or right from a point, while in higher dimensions, one can move in many directions. Thus, if the derivative does not vanish, one must argue that there is ''some'' direction in which the function increases – and thus in the opposite direction the function decreases. This is the only change to the proof or the analysis.

The statement can also be extended to [[differentiable manifolds]]. If <math>f : M \to \mathbb{R}</math> is a [[Differentiable manifold#Differentiable functions|differentiable function]] on a manifold <math>M</math>, then its local extrema must be [[Critical point (mathematics)|critical points]] of <math>f</math>, in particular points where the [[exterior derivative]] <math>df</math> is zero.<ref>{{cite web|title=Is Fermat's theorem about local extrema true for smooth manifolds? |date=August 11, 2015 |url=https://math.stackexchange.com/q/1392981 |work=[[Stack Exchange]] |accessdate=21 April 2017}}</ref>{{better source|date=January 2023}}

==Applications==
{{See also|Maxima and minima}}
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the [[Zero of a function|zeros]] of the derivative), the non-differentiable points, and the boundary points, and then investigating this [[set (mathematics)|set]] to determine the extrema.

One can do this either by evaluating the function at each point and taking the maximum, or by analyzing the derivatives further, using the [[first derivative test]], the [[second derivative test]], or the [[higher-order derivative test]].

==Intuitive argument==
Intuitively, a differentiable function is approximated by its derivative – a differentiable function behaves infinitesimally like a [[linear function]] <math>a+bx,</math> or more precisely, <math>f(x_0) + f'(x_0)(x-x_0).</math> Thus, from the perspective that "if ''f'' is differentiable and has non-vanishing derivative at <math>x_0,</math> then it does not attain an extremum at <math>x_0,</math>" the intuition is that if the derivative at <math>x_0</math> is positive, the function is ''[[monotonic function|increasing]]'' near <math>x_0,</math> while if the derivative is negative, the function is ''[[monotonic function|decreasing]]'' near <math>x_0.</math> In both cases, it cannot attain a maximum or minimum, because its value is changing. It can only attain a maximum or minimum if it "stops" – if the derivative vanishes (or if it is not differentiable, or if one runs into the boundary and cannot continue). However, making "behaves like a linear function" precise requires careful analytic proof.

More precisely, the intuition can be stated as: if the derivative is positive, there is ''some point'' to the right of <math>x_0</math> where ''f'' is greater, and ''some point'' to the left of <math>x_0</math> where ''f'' is less, and thus ''f'' attains neither a maximum nor a minimum at <math>x_0.</math> Conversely, if the derivative is negative, there is a point to the right which is lesser, and a point to the left which is greater. Stated this way, the proof is just translating this into equations and verifying "how much greater or less".

The [[Intuition (knowledge)|intuition]] is based on the behavior of [[polynomial function]]s. Assume that function ''f'' has a maximum at ''x''<sub>0</sub>, the reasoning being similar for a function minimum. If <math>x_0 \in (a,b)</math> is a local maximum then, roughly, there is a (possibly small) [[neighbourhood (mathematics)|neighborhood]] of <math>x_0</math> such as the function "is increasing before" and "decreasing after"<ref group="note">This intuition is only correct for [[continuously differentiable]] <math>\left(C^1\right)</math> functions, while in general it is not literally correct&mdash;a function need not be increasing up to a local maximum: it may instead be oscillating, so neither increasing nor decreasing, but simply the local maximum is greater than any values in a small neighborhood to the left or right of it. See details in the pathologies.</ref> <math>x_0</math>. As the derivative is positive for an increasing function and negative for a decreasing function, <math>f'</math> is positive before and negative after <math>x_0</math>. <math>f'</math> does not skip values (by [[Darboux's theorem (analysis)|Darboux's theorem]]), so it has to be zero at some point between the positive and negative values. The only point in the neighbourhood where it is possible to have <math>f'(x) = 0</math> is <math>x_0</math>.

The theorem (and its proof below) is more general than the intuition in that it does not require the function to be differentiable over a neighbourhood around <math>x_0</math>. It is sufficient for the function to be differentiable only in the extreme point.

==Proof==

=== Proof 1: Non-vanishing derivatives implies not extremum ===
Suppose that ''f'' is differentiable at <math>x_0 \in (a,b),</math> with derivative ''K,'' and assume [[without loss of generality]] that <math>K > 0,</math> so the tangent line at <math>x_0</math> has positive slope (is increasing). Then there is a neighborhood of <math>x_0</math> on which the [[secant line]]s through <math>x_0</math> all have positive slope, and thus to the right of <math>x_0,</math> ''f'' is greater, and to the left of <math>x_0,</math> ''f'' is lesser.

The schematic of the proof is:
* an infinitesimal statement about derivative (tangent line) ''at'' <math>x_0</math> implies
* a local statement about difference quotients (secant lines) ''near'' <math>x_0,</math> which implies
* a local statement about the ''value'' of ''f'' near <math>x_0.</math>

Formally, by the definition of derivative, <math>f'(x_0) = K</math> means that
:<math>\lim_{\varepsilon \to 0} \frac{f(x_0+\varepsilon)-f(x_0)}{\varepsilon} = K.</math>
In particular, for sufficiently small <math>\varepsilon</math> (less than some <math>\varepsilon_0</math>), the quotient must be at least <math>K/2,</math> by the definition of limit. Thus on the [[Interval (mathematics)|interval]] <math>(x_0-\varepsilon_0,x_0+\varepsilon_0)</math> one has:
:<math>\frac{f(x_0+\varepsilon)-f(x_0)}{\varepsilon} > K/2;</math>
one has replaced the ''equality'' in the limit (an infinitesimal statement) with an ''inequality'' on a neighborhood (a local statement). Thus, rearranging the equation, if <math>\varepsilon > 0,</math> then:
:<math>f(x_0+\varepsilon) > f(x_0) + (K/2)\varepsilon > f(x_0),</math>
so on the interval to the right, ''f'' is greater than <math>f(x_0),</math> and if <math>\varepsilon < 0,</math> then:
:<math>f(x_0+\varepsilon) < f(x_0) + (K/2)\varepsilon < f(x_0),</math>
so on the interval to the left, ''f'' is less than <math>f(x_0).</math>

Thus <math>x_0</math> is not a local or global maximum or minimum of ''f.''

=== Proof 2: Extremum implies derivative vanishes ===
Suppose that <math>x_0</math> is a local maximum.  (A similar argument applies if <math>x_0</math> is a local minimum.) Then there is some [[Neighbourhood (mathematics)|neighbourhood]] around <math>x_0</math> such that <math>f(x_0) \ge f(x)</math> for all <math>x</math> within that neighborhood. If <math>x > x_0</math>, then the [[difference quotient]] <math>\frac{f(x) - f(x_0)}{x - x_0}</math> is non-positive for <math>x</math> in this neighborhood. This implies
<math display="block">
\lim_{x\rightarrow x_0^+}\frac{f(x) - f(x_0)}{x - x_0} \le 0.
</math>
Similarly, if <math>x < x_0</math>, then the difference quotient is non-negative, and so
<math display="block">
\lim_{x\rightarrow x_0^-}\frac{f(x) - f(x_0)}{x - x_0} \geq 0.
</math>
Since <math>f</math> is differentiable, the above limits must both be equal to <math>f'(x_0)</math>. This is only possible if both limits are equal to 0, so <math>f'(x_0) = 0</math>.<ref>{{Cite book |last=Canuto |first=Claudio |url=https://link.springer.com/book/10.1007/978-3-319-12772-9 |title=Mathematical Analysis I |last2=Tabacco |first2=Anita |publisher=[[Springer Cham]] |year=2015 |isbn=978-3-319-12771-2 |edition=2nd |doi=10.1007/978-3-319-12772-9}}</ref>{{Rp|page=182}}

== Cautions ==
A subtle misconception that is often held in the context of Fermat's theorem is to assume that it makes a stronger statement about local behavior than it does. Notably, Fermat's theorem does ''not'' say that functions (monotonically) "increase up to" or "decrease down from" a local maximum. This is very similar to the misconception that a limit means "monotonically getting closer to a point". For "well-behaved functions" (which here means [[continuously differentiable]]), some intuitions hold, but in general functions may be ill-behaved, as illustrated below. The moral is that derivatives determine ''infinitesimal'' behavior, and that ''[[Continuous function|continuous]]'' derivatives determine ''local'' behavior.

=== Continuously differentiable functions ===
If ''f'' is [[continuously differentiable]] <math>\left(C^1\right)</math> on an [[open neighborhood]] of the point <math>x_0</math>, then <math>f'(x_0) > 0</math> does mean that ''f'' is increasing on a neighborhood of <math>x_0,</math> as follows.

If <math>f'(x_0) = K > 0</math> and <math>f \in C^1,</math> then by continuity of the derivative, there is some <math>\varepsilon_0 > 0</math> such that <math>f'(x) > K/2</math> for all <math>x \in (x_0 - \varepsilon_0, x_0 + \varepsilon_0)</math>. Then ''f'' is increasing on this interval, by the [[mean value theorem]]: the slope of any secant line is at least <math>K/2,</math> as it equals the slope of some tangent line.

However, in the general statement of Fermat's theorem, where one is only given that the derivative ''at'' <math>x_0</math> is positive, one can only conclude that secant lines ''through'' <math>x_0</math> will have positive slope, for secant lines between <math>x_0</math> and near enough points.

Conversely, if the derivative of ''f'' at a point is zero (<math>x_0</math> is a stationary point), one cannot in general conclude anything about the local behavior of ''f'' – it may increase to one side and decrease to the other (as in <math>x^3</math>), increase to both sides (as in <math>x^4</math>), decrease to both sides (as in <math>-x^4</math>), or behave in more complicated ways, such as oscillating (as in <math>x^2 \sin(1/x)</math>, as discussed below).

One can analyze the infinitesimal behavior via the [[second derivative test]] and [[higher-order derivative test]], if the function is differentiable enough, and if the first non-vanishing derivative at <math>x_0</math> is a [[continuous function]], one can then conclude local behavior (i.e., if <math>f^{(k)}(x_0) \neq 0</math> is the first non-vanishing derivative, and <math>f^{(k)}</math> is continuous, so <math>f \in C^k</math>), then one can treat ''f'' as locally close to a polynomial of [[Degree of a polynomial|degree]] ''k,'' since it behaves approximately as <math>f^{(k)}(x_0) (x - x_0)^k,</math> but if the ''k''-th derivative is not continuous, one cannot draw such conclusions, and it may behave rather differently.

=== Pathological functions ===
The function <math>\sin(1/x)</math> oscillates increasingly rapidly between <math>-1</math> and <math>1</math> as ''x'' approaches 0. Consequently, the function <math>f(x) = (1 + \sin(1/x))x^2</math> oscillates increasingly rapidly between 0 and <math>2x^2</math> as ''x'' approaches 0. If one extends this function by defining <math>f(0) = 0</math> then the extended function is continuous and everywhere differentiable (it is differentiable at 0 with derivative 0), but has rather unexpected behavior near 0: in any neighborhood of 0 it attains 0 infinitely many times, but also equals <math>2x^2</math> (a positive number) infinitely often.

Continuing in this vein, one may define <math>g(x) = (2 + \sin(1/x))x^2</math>, which oscillates between <math>x^2</math> and <math>3x^2</math>.  The function has its local and global minimum at <math>x=0</math>, but on no neighborhood of 0 is it decreasing down to or increasing up from 0 – it oscillates wildly near 0.

This pathology can be understood because, while the function {{mvar|g}} is everywhere differentiable, it is not ''continuously'' differentiable: the limit of <math>g'(x)</math> as <math>x \to 0</math> does not exist, so the derivative is not continuous at 0. This reflects the oscillation between increasing and decreasing values as it approaches 0.

== See also ==
* [[Optimization (mathematics)]]
* [[Maxima and minima]]
* [[Derivative]]
* [[Extreme value]]
* [[arg max]] <!-- lower case intentional -->
* [[Adequality]]

== Notes ==
{{Reflist|group=note}}

== References ==
{{Reflist}}

==External links==
* {{PlanetMath reference|urlname=FermatsTheoremStationaryPoints|title=Fermat's Theorem (stationary points)}}
* {{PlanetMath reference|urlname=ProofOfFermatsTheoremStationaryPoints|title=Proof of Fermat's Theorem (stationary points)}}


{{Pierre de Fermat}}

{{DEFAULTSORT:Fermat's Theorem (Stationary Points)}}
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