Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. It gives a method to find local maxima and minima of continuous functions by showing that every local extremum of the function is a stationary point (the function derivative is zero in that point). So, by using Fermat's theorem, the problem of finding a function extremum is reduced to solving an equation. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Stationary points (red pluses) and inflection points (green circles). ... In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...

It is important to note that Fermat's theorem gives only a necessary condition for extreme function values. That is, some stationary points are not extreme values, they are inflection points. To check if a stationary point is an extreme value and to further distinguish between a function maximum and a function minimum it is necessary to analyse the second derivative (if it exists). In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... Plot of y = x3 with inflection point of (0,0). ...

Fermat's theorem

Let f : (a,b) → R be a continuous function and suppose that x₀ in (a,b) is a local extremum of f. If f is differentiable at x₀ then fˈ(x₀) = 0. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

Intuition

We give the intuition for a function maximum, the reasoning being similar for a function minimum. If x₀ in (a,b) is a local maximum then there is a (possibly small) neighborhood of x₀ such as the function is increasing before and decreasing after x₀. As the derivative is positive for an increasing function and negative for a decreasing function, fˈ is positive before and negative after x₀. fˈ doesn't skip values (by 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 fˈ(x) = 0 is x₀. In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

Note that the theorem (and its proof below) is more general than the intuition in that it doesn't require the function to be differentiable over a neighbourhood around x₀. As stated in the theorem, it is sufficient for the function to be differentiable only in the extreme point.

Proof

Suppose that x₀ is a local maximum (a similar proof applies if x₀ is a local minimum). Then there exists δ > 0 such that (x₀ - δ,x₀ + δ) is a subset of (a,b) and such that we have f(x₀) ≥ f(x) for all x with |x - x₀| < δ. Hence for any h in (0,δ) we notice that it holds

Since the limit of this ratio as h → 0 from above exists and is equal to fˈ(x₀) we conclude that fˈ(x₀) ≤ 0. On the other hand for h in (-δ,0) we notice that In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

but again the limit as h → 0 from below exists and is equal to fˈ(x₀) so we also have fˈ(x₀) ≥ 0.

Hence we conclude that fˈ(x₀) = 0.

See also

• Derivative
• Extreme value
• Stationary point
• Inflection point
• Pierre de Fermat

In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... Stationary points (red pluses) and inflection points (green circles). ... Plot of y = x3 with inflection point of (0,0). ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ...

External links

• This article incorporates material from Fermat's Theorem (stationary points) on PlanetMath, which is licensed under the GFDL.
• This article incorporates material from Proof of Fermat's Theorem (stationary points) on PlanetMath, which is licensed under the GFDL.