Fermat's theorem (stationary points)


In mathematics, Fermat's theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point. Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat.
By using Fermat's theorem, the potential extrema of a function, with derivative, are found by solving an equation in. Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points. The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.

Statement

One way to state Fermat's 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:
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

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:
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 is a differentiable function on a manifold, then its local extrema must be critical points of, in particular points where the exterior derivative is zero.

Applications

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, the non-differentiable points, and the boundary points, and then investigating this 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 or more precisely, Thus, from the perspective that "if f is differentiable and has non-vanishing derivative at then it does not attain an extremum at " the intuition is that if the derivative at is positive, the function is increasing near while if the derivative is negative, the function is decreasing near 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. 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 where f is greater, and some point to the left of where f is less, and thus f attains neither a maximum nor a minimum at 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 is based on the behavior of polynomial functions. Assume that function f has a maximum at x0, the reasoning being similar for a function minimum. If is a local maximum then, roughly, there is a neighborhood of such as the function "is increasing before" and "decreasing after". As the derivative is positive for an increasing function and negative for a decreasing function, is positive before and negative after. doesn't skip values, 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 is.
The theorem is more general than the intuition in that it doesn't require the function to be differentiable over a neighbourhood around. 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 with derivative K, and assume without loss of generality that so the tangent line at has positive slope. Then there is a neighborhood of on which the secant lines through all have positive slope, and thus to the right of f is greater, and to the left of f is lesser.
The schematic of the proof is:
Formally, by the definition of derivative, means that
In particular, for sufficiently small , the fraction must be at least by the definition of limit. Thus on the interval one has:
one has replaced the equality in the limit with an inequality on a neighborhood. Thus, rearranging the equation, if then:
so on the interval to the right, f is greater than and if then:
so on the interval to the left, f is less than
Thus is not a local or global maximum or minimum of f.

Proof 2: Extremum implies derivative vanishes

Alternatively, one can start by assuming that is a local maximum, and then prove that the derivative is 0.
Suppose that is a local maximum. Then there such that and such that we have with. Hence for any we notice that it holds
Since the limit of this ratio as gets close to 0 from above exists and is equal to we conclude that. On the other hand, for we notice that
but again the limit as gets close to 0 from below exists and is equal to so we also have.
Hence we conclude that

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 "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", 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 derivatives determine local behavior.

Continuously differentiable functions

If f is continuously differentiable on an open neighborhood of the point, then does mean that f is increasing on a neighborhood of as follows.
If and then by continuity of the derivative, there is some such that . Then f is increasing on this interval, by the mean value theorem: the slope of any secant line is at least 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 is positive, one can only conclude that secant lines through will have positive slope, for secant lines between and near enough points.
Conversely, if the derivative of f at a point is zero, one cannot in general conclude anything about the local behavior of f – it may increase to one side and decrease to the other, increase to both sides, decrease to both sides, or behave in more complicated ways, such as oscillating.
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 is a continuous function, one can then conclude local behavior, then one can treat f as locally close to a polynomial of degree k, since it behaves approximately as but if the k-th derivative is not continuous, one cannot draw such conclusions, and it may behave rather differently.

Pathological functions

The function – it oscillates increasingly rapidly between and as x approaches 0. Consequently, the function oscillates increasingly rapidly between 0 and as x approaches 0. If one extends this function by defining then the extended function is continuous and everywhere differentiable, but has rather unexpected behavior near 0: in any neighborhood of 0 it attains 0 infinitely many times, but also equals infinitely often.
Continuing in this vein, one may define, which oscillates between and. The function has its local and global minimum at, 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 is everywhere differentiable, it is not continuously differentiable: the limit of as does not exist, so the derivative is not continuous at 0. This reflects the oscillation between increasing and decreasing values as it approaches 0.